\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 01, pp. 1--24.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/01\hfil Resonant $(p,q)$-equations]
{Resonant $(p,q)$-equations with Robin \\ boundary condition}

\author[M. E. Filippakis, N. S. Papageorgiou \hfil EJDE-2018/01\hfilneg]
{Michael E. Filippakis, Nikolaos S. Papageorgiou}

\address{Michael E. Filippakis \newline
Department of Digital Systems,
University of Piraeus,
Piraeus 18536, Greece}
\email{mfilip@unipi.gr}

\address{Nikolaos S. Papageorgiou \newline
Department of Mathematics,
National Technical University of Athens,
Zografou Campus, Athens 15780, Greece}
\email{npapg@math.ntua.gr}

\dedicatory{Communicated by Mitsuharu Otani}

\thanks{Submitted October 4, 2017. Published January 2, 2018.}
\subjclass[2010]{35J20, 35J92, 58E05}
\keywords{$(p,q)$-Laplace differential operator;  resonance; critical groups; 
\hfill\break\indent multiple solutions; nonlinear regularity}

\begin{abstract}
 We consider a nonlinear nonhomogeneous Robin problem that has the
 sum of a $p$-Laplacian and a $q$-Laplacian (a $(p,q)$-equation).
 The reaction term is a Caratheodory function which is resonant at
 $\pm\infty$ with respect to any nonprincipal variational eigenvalue
 of the Robin $p$-Laplacian. Using variational methods and Morse
 theory (critical groups), we show the existence of at least
 three nontrivial smooth solutions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction} \label{Intr}

Let $\Omega\subseteq\mathbb{R^N}$ be a bounded domain with a $C^2$ boundary
$\partial \Omega$. In this paper we study the following nonlinear
nonhomogeneous elliptic equation
\begin{equation}\label{eq1}
\begin{gathered}
 -\Delta_p u(z)-\Delta_q u(z)=f(z,u(z))\quad\text{in }\Omega, \\
 \frac{\partial u}{\partial n_{pq}}+\beta_1 (z)|u|^{p-2}u
 +\beta_2(z)|u|^{q-2}u=0 \quad\text{on }\partial \Omega.
\end{gathered}
 \end{equation}
Here $1<q<p<+\infty$, and for $1<r<\infty$ by $\Delta_r$ we denote the
$r$-Laplace differential operator
$$
\Delta_r u=\operatorname{div}(|Du|^{r-2}Du)\quad\text{for all }
u\in W^{1,r}(\Omega).
$$

The reaction term $f(z,x)$ is a Caratheodory function (that is, for all
$x\in \mathbb{R}$ $z\to f(z,x)$ is measurable and for a.a. $z\in \Omega$
 $x\to f(z,x)$ is continuous). We assume that $f(z,\cdot)$ exhibits
$(p-1)$-linear growth near $\pm\infty$ and interacts with the non-principal
part of the variational spectrum of the Robin $p$-Laplacian (resonant problem).
In the boundary condition $\frac{\partial u}{\partial n_{pq}}$ denotes the
conormal derivative of $u$ defined by extension of the map
$$
C^1(\overline{\Omega})\ni u\to \Big(|Du|^{p-2}+|Du|^{q-2}\Big)
\frac{\partial u}{\partial n},
$$
with $n(\cdot)$ being the outward unit normal on $\partial \Omega$.
This generalized normal derivative is dictated by the nonlinear Green's
identity (see Gasinski-Papageorgiou \cite[p.210]{GPap1}).
 It is also used by Lieberman \cite{aLieberman1991}.

In this article combining variational methods based on the critical
point theory and Morse theory (critical groups), we prove a multiplicity
theorem, showing the existence of at least three nontrivial smooth
solutions. Note that the differential operator
$u\to \Delta_p u+\Delta_q u$ is nonhomogeneous.

Equations driven by the sum of a $p$-Laplacian and of a $q$-Laplacian
are known as $(p,q)$-equations and arise in problems of mathematical physics.
We refer to Benci-D'Avenia-Fortunato-Pisani \cite{BenAvenFortPis}
(quantum physics) and Cherfils-Ilyason \cite{CherfilsILyason} (plasma physics).
There have been existence and multiplicity results for such equations.
 We mention the works of Aizicovici-Papageorgiou-Staicu \cite{AizicPapStaicu2},
Benouhiba-Belyacine \cite{BenBel}, Bobkov-Tanaka \cite{BobkovTanaka},
Cingolani-Degiovanni \cite{CingolaniDegiovanni},
Gasinski-Papageorgiou \cite{GPap2},
Marano-Mosconi-Papageorgiou \cite{MaranoMosconiPap},
Marano-Papageorgiou \cite{MaranoPap}, Mugnai-Papageorgiou \cite{MugnaiPap2},
Papageorgiou-Radulescu \cite{PaRa1}, \cite{PaRa2},
Papageorgiou-Winkert \cite{PaWinkert}, Sun \cite{Sun},
Sun-Zhang-Su \cite{SunZhangSu}, Tanaka \cite{Tanaka},
Yang-Yin \cite{YangYin} (Dirichlet problems) and
Papageorgiou-Radu\-lescu \cite{PaRa3} (Neumann and Robin problems).
From the aforementioned papers resonant problems are examined in
 Gasinski-Papageorgiou \cite{GPap2}, Papageorgiou-Radulescu \cite{PaRa1}
 and Sun \cite{Sun} and the resonance is with respect to the principal
eigenvalue of $(-\Delta_p$, $ W_0^{1,p}(\Omega))$.

\section{Preliminary results and hypotheses}

Let $X$ be a Banach space and $X^*$ its topological dual.
By $\langle \cdot, \cdot \rangle$, we denote the duality brackets for
the pair $(X,X^*)$.

Let $\varphi\in C^1(X,\mathbb{R})$.
 We say that $\varphi$ satisfies the ``Cerami condition''
 (the ``C-condition'' for short), if the following property holds:
``every sequence $\{u_n\}_{n\geq 1}\subseteq X$ such that
$\{\varphi(u_n)\}_{n\in \mathbb{N}}\subseteq \mathbb{R}$ is bounded and
$(1+\|u_n\|)\varphi'(u_n)\to 0$ in $X^*$ as $n\to \infty$, admits a
strongly convergent subsequence''.

This is a compactness type condition on the functional $\varphi$.
It leads to a deformation lemma, from which one can derive the minimax
theory for the critical values of $\varphi$. Prominent in that theory,
is the so-called ``mountain pass theorem'' of Ambrosetti-Rabinowitz \cite{AmbRab}.
 Here we state the result in a slightly more general form
(see Gasinski-Papageorgiou \cite[p.648]{GPap1}).

\begin{theorem}\label{thm1}
If $\varphi\in C^1(X)$ satisfies the $C$-condition, $u_0,u_1\in X$, $\rho>0$,
$\|u_1-u_0\|>\rho$,
$$
\max\{\varphi(u_0),\varphi(u_1)\}<\inf[\varphi(u):\|u-u_0\|=\rho]=m_{\rho}
$$
and $c=\inf_{\gamma\in \Gamma}\max_{0\leq t\leq 1}\varphi(\gamma(t))$
where $\Gamma=\{\gamma\in C([0,1],X):\gamma(0)=u_0,\gamma(1)=u_1\}$,
then $c\geq m_{\rho}$ and $c$ is a critical value of $\varphi$ (i.e.,
there exists $u^*\in X$ such that $\varphi'(u^*)=0$ and $\varphi(u^*)=c$ in $X^*$).
\end{theorem}

For the analysis of problem \eqref{eq1} we will use the Sobolev spaces
$W^{1,r}(\Omega)$ $1<r<\infty$. We know that this is a Banach space with norm
\[
\|u\|=\big[\|u\|_r^r+\|Du\|_r^r\big]^{1/r}\quad\text{for all }
u\in W^{1,r}(\Omega).
\]
The Banach space $W^{1,r}(\Omega)$ is uniformly convex, thus reflexive.

We will also use the subspace $C^1(\overline{\Omega})$.
We will exploit the fact that $C^1(\overline{\Omega})$ is an ordered
Banach space with positive (order) cone
$$
C_+=\{u\in C^1(\overline{\Omega}):u(z)\geq 0\quad\text{for all }z\in \overline{\Omega}\}$$

This cone has a nonempty interior containing the set
$$
D_+=\{u\in C_+:u(z)>0\quad \text{for all }z\in \overline{\Omega}\}
$$
On $\partial \Omega$ we consider the $(N-1)$-dimensional Hausdorff (surface)
 measure $\sigma (\cdot)$. Using this measure on $\partial\Omega$, we can
define in the usual way the boundary Lebesgue spaces
$L^{\tau}(\partial\Omega)$ $1\leq\tau \leq\infty$.

From the theory of Sobolev spaces, we know that there exists a unique continuous
linear map $\gamma_0:W^{1,\tau}(\Omega)\to L^{\tau}(\partial \Omega)$,
 known as the ``trace map'' such that
$$
\gamma_0(u)=u|_{\partial \Omega}\quad \text{for all }
u\in W^{1,\tau}(\Omega)\cap C(\overline{\Omega}).
$$
The trace map assigns boundary values to any Sobolev function.
We know that $\gamma_0(\cdot)$ is compact into
$L^{\theta}(\partial \Omega)$ for all $\theta\in [1,\frac{(N-1)p}{N-p})$
if $\tau<N$ and into $L^{\theta}(\partial \Omega)$ for all
$\theta\in [1,+\infty)$ if $\tau\geq N$. Moreover, we have
$$
\operatorname{im}\gamma_0=W^{\frac{1}{\tau'},\tau}(\partial \Omega)\quad
(\frac{1}{\tau}+\frac{1}{\tau'}=1)\text{ and }\ker\gamma_0=W_0^{1,\tau}(\Omega).
$$

In what follows, for the sake of notational simplicity, we drop the use of the
trace map $\gamma_0$. All restrictions of Sobolev functions on $\partial \Omega$,
are understood in the sense of traces.
For $1<r<\infty$, let $A_r:W^{1,r}(\Omega)\to W^{1,r}(\Omega)^{*}$ be defined by
$$
\langle A_{r}(u),h\rangle =\int_{\Omega} |Du|^{r-2}(Du,Dh)_{\mathbb{R^N}}\,dz\quad
\text{for all }u,h\in W^{1,r}(\Omega).
$$
For this operator we have the following result
(see Motreanu-Motreanu-Papageorgiou \cite[p.40]{MotreanuMotreanuPap}).

\begin{proposition}\label{prop2}
The map $A_r:W^{1,r}(\Omega)\to W^{1,r}(\Omega)^{*}$ is bounded
(that is, maps bounded sets to bounded sets), continuous, monotone
(hence maximal monotone too) and of type ($S)_+$, that is
$$
u_n\stackrel{w} \to u\quad\text{in }W^{1,r}(\Omega)\quad\text{and}\quad
\limsup_{n\to \infty}\langle A_r (u_n),u_n-u\rangle \leq 0
\Rightarrow u_n\to u \text{ in }W^{1,r}(\Omega).
$$
\end{proposition}

Let $f_0:\Omega\times X\to \mathbb{R}$ be a Caratheodory function such that
$$
|f_0(z,x)|\leq \alpha_0(z)[1+|x|^{\tau-1}]\quad\text{for a.a.
$z\in \Omega$, all }x\in \mathbb{R},
$$
with $\alpha_0\in L^{\infty}(\Omega)_+=\{\alpha\in L^{\infty}(\Omega):
\alpha(z)\geq 0\text{ for a.a. } z\in \Omega\}$ and $\tau\in (1,p^*]$ where
\[
p^*=  \begin{cases}
\frac{Np}{N-p} & \text{if } p<N\\
 +\infty & \text{if } N\leq p
\end{cases}
\]
(the critical Sobolev exponent for $p$). Also, let
 $k_0\in C^{0,\eta}(\partial \Omega\times \mathbb{R})$ with $\eta\in (0,1)$ and
$$
0\leq k_0(z,x)\leq c_1(|x|^q+|x|^p)\quad\text{for all }
(z,x)\in \partial \Omega \times \mathbb{R},
$$
with $c_1>0$, $1<q<p$. We set
$$
F_0(z,x)=\int_0^xf_0(z,s) ds\quad\text{and}\quad K_0(z,x)=\int_0^xk_0(z,s) ds
$$
and consider the $C^1$-functional $\varphi_0:W^{1,p}(\Omega)\to \mathbb{R}$
defined by
$$
\varphi_0(u)=\frac{1}{p}\|Du\|_p^p+\frac{1}{q}\|Du\|_q^q
+\int_{\partial \Omega} K_0(z,u) d \sigma -\int_{\Omega}F_0(z,u)\,dz\quad
\text{for all } u\in W^{1,p}(\Omega).
$$
The next proposition is a special case of a more general result of
Papageorgiou-Radulescu \cite{PaRa5}.

\begin{proposition}\label{prop3}
If $u_0\in W^{1,p}(\Omega)$ is a local $C^1(\overline{\Omega})$-minimizer of
$\varphi_0$, that is, there exists $\rho_0>0$ such that
$$
\varphi_0(u_0)\leq \varphi_0(u_0+h)\quad\text{for all }
h\in C^1(\overline{\Omega})\text{ with }\|h\|_{C^1(\overline{\Omega})}\leq \rho_0,
$$
then $u_0\in C^1(\overline{\Omega})$ and $u_0$ is a local $W^{1,p}(\Omega)$-minimizer
of $\varphi_0$, that is, there exists $\rho_1>0$ such that
$$
\varphi_0(u_0)\leq \varphi_0(u_0+h)\quad\text{for all }
h\in W^{1,p}(\Omega)\text{ with }\|h\|\leq \rho_1.
$$
\end{proposition}

As we already mentioned in the introduction, one of our tools, are critical groups.
So, let us recall their definition.
Let $X$ be a Banach space, $\varphi\in C^1(X,\mathbb{R})$ and
$c\in \mathbb{R}$. We introduce the following sets:
\begin{gather*}
K_{\varphi}=\{u\in X:\varphi'(u)=0\},\quad
K_{\varphi}^c=\{u\in K_{\varphi}:\varphi(u)=c\},\\
\varphi^c=\{u\in X:\varphi(u)\leq c\}.
\end{gather*}

Suppose that $(Y_1,Y_2)$ is a topological pair such that
$Y_2\subseteq Y_1\subseteq X$ and $k\in \mathbb{N}_0$. By $H_k(Y_1,Y_2)$
we denote the kth-relative singular homology group with integer coefficients
for the pair $(Y_1,Y_2)$. Let $u\in K_{\varphi}^{c}$ be isolated.
The critical groups of $\varphi$ at $u$ are defined by
$$
C_k(\varphi,u)=H_k(\varphi^c\cap U,\varphi^c\cap U\backslash\{u\})\text{ for all }
k\in \mathbb{N}_0,
$$
where $U$ is an isolating neighborhood of $u$, that is,
$K_{\varphi}\cap \varphi^c \cap U=\{u\}$. The excision property of singular
homology, implies that the above definition of critical groups is
independent of the particular choice of the isolating neighborhood $U$.

Suppose that $\varphi$ satisfies the $C$-condition and assume that
$\inf\varphi(K_{\varphi})>-\infty$. Let $c<\inf\varphi(K_{\varphi})$.
The critical groups of $\varphi$ at infinity are defined by
$$
C_k(\varphi,\infty)=H_k(X,\varphi^{c})\quad\text{for all }k\in \mathbb{N}_0.
$$
This definition is independent of the choice of the level
$c<\inf\varphi(K_{\varphi})$. Indeed, let $c'<c<\inf\varphi(K_{\varphi})$.
From  Motreanu-Motreanu-Papageorgiou
\cite[Corollary 5.3.5, p.115]{MotreanuMotreanuPap}, we have that:
if $\varphi^{c'}$ is a strong deformation retract of $\varphi^{c}$, then
\[
H_k(X,\varphi^{c})=H_k(X,\varphi^{c'})\quad \text{for all }k\in \mathbb{N}_0
\]
(see  Motreanu-Motreanu-Papageorgiou
\cite[Corollary 6.15, p.145 of]{MotreanuMotreanuPap}).

The next result is a useful tool for computing the critical groups at infinity.
It extends an earlier analogous result for Hilbert spaces of Liang-Su \cite{LiangSu}.

\begin{proposition}\label{prop4}
If $X$ is a Banach space, $(t,u)\to h_t (u)$ belongs in
$C^1([0,1]\times X, \mathbb{R})$, maps bounded sets to bounded sets,
the maps $u\to (h_t)'(u)$ and $t\to \partial _t h_t(u)$ are both locally Lipschitz,
$h_0,h_1$ satisfy the C-condition
$$
|\partial_t h_t(u)|\leq c_2(\|u\|^q+\|u\|^p)\quad \text{for all }u\in X,
$$
with $c_2>0$, $1<q<p<\infty$ and there exist $\theta_0\in \mathbb{R}$ and
$\delta_0>0$ such that
$$
h_t(u)\leq \theta_0\Rightarrow (1+\|u\|)\|(h_t)'(u)\|_{*}
\geq \delta_0[\|u\|^q+\|u\|^p]\quad \text{for all }t\in [0,1],
$$
then $C_k(h_0,\infty)=C_k(h_1,\infty)$ for all $k\in \mathbb{N}_0$.
\end{proposition}

\begin{proof}
Since $h\in C^1([0,1]\times X,\mathbb{R})$, it admits a pseudogradient vector
field $\widehat{v}_t(u)$ (see Gasinski-Papageorgiou
\cite[Theorem 5.1.19, p.616]{GPap1}).
In fact from the construction of the pseudogradient vector field, we have
$$
\widehat{v}_t(u)=(\partial_t h_t(u),v_t(u)),
$$
with $v_t(\cdot)$ being a pseudogradient vector field corresponding to the
function $h_t(\cdot)$. Therefore, for all $t\in [0,1]$ and all
$u\in X\backslash {K_{h_t}}$, we have
\begin{equation}\label{eq2}
\|(h_t)'(u)\|^2_{*}\leq \langle (h_t)'(u),v_t(u)\rangle\quad\text{and}\quad
\|v_t(u)\|\leq 2\|(h_t)'(u)\|_{*}
\end{equation}

For $t\in [0,1]$ we consider the vector field $g_t:X\backslash {K_{h_t}}\to X$
defined by
\begin{equation}\label{eq3}
g_t(u)=-\frac{|\partial_t h_t(u)|}{\|(h_t)'(u)\|^2_{*}} v_t(u)\quad
\text{for all }u\in X\backslash {K_{h_t}}.
\end{equation}
This is a locally Lipschitz vector field. Choose $\theta\leq \theta_0$ such that
$$
h_0^{\theta}\neq \emptyset\quad\text{or}\quad h_1^{\theta}\neq \emptyset
$$
If no such $\theta\leq \theta_0$ can be found, it means that both
$h_0,h_1$ are bounded below. Since by hypothesis they satisfy the C-condition,
we have
$$
C_k(h_0,\infty)=C_k(h_1,\infty)=\delta_{k,0}\mathbb{Z}\quad
\text{for all } k\in \mathbb{N}_0
$$
(see  Motreanu-Motreanu-Papageorgiou
\cite[Proposition 6.64, p.161]{MotreanuMotreanuPap}).
 Hence the conclusion of the proposition holds.

To fix things, we assume that $h_0^{\theta}\neq \emptyset$
(the reasoning is similar if $h_1^{\theta}\neq \emptyset$).
Let $u\in h_0^{\theta}$ and consider the  abstract Cauchy problem
\begin{equation}\label{eq4}
\frac{d\xi}{dt}=g_t(\xi)\quad\text{on }[0,1],\;\tau(0)=u.
\end{equation}

Since the vector field is locally Lipschitz, problem \eqref{eq4} admits a
local flow (see Gasinski-Papageorgiou \cite[Theorem 5.1.21, p.618]{GPap1}).
We denote this local flow by $\xi(t,u)$. For the sake of notational simplicity,
in the next calculation we drop the $u$-dependence in the expression of
the local flow, since it does not play any role. We have
\begin{align*}
&\frac{d}{dt}h_t(\xi(t)) \\
&=\langle (h_t)'(\xi(t)),\frac{d \xi}{dt}\rangle+\partial_t h_t(\xi(t))\\
&=\langle (h_t)'(\xi(t)),-\frac{|\partial_t h_t(\xi(t))|}{\|(h_t)'(\xi(t))
\|_{*}} v_t(\xi(t))\rangle+\partial_t h_t(\xi(t))\quad
\text{(see \eqref{eq3} and\eqref{eq4})}\\
&\leq -|\partial_t h_t(\xi(t))|+\partial_t h_t(\xi(t))\;\;\text{(see \eqref{eq2})}\\
&\leq 0.
\end{align*}
Hence for $t>0$ small, we have
\[
h_t(\xi(t))\leq h_0(\xi(0))=h_0(u)\leq \theta\leq \theta_0
\]
(see \eqref{eq4} and recall that $u\in h_0^{\theta}$), which implies
\begin{equation} \label{eq5}
(1+\|\xi(t)\|)\|(h_t)'(\xi(t))\|_{*}\geq \delta_0\big[\|\xi(t))\|^q+\|\xi(t)\|^p\big].
\end{equation}
Therefore,
\begin{align*}
\|g_t(\xi(t))\|
&\leq \frac{|\partial_t h_t(\xi(t))|}{\|(h_t)'(\xi(t))\|^2_{*}}\|v_t(\xi(t))\|
\quad \text{(see \eqref{eq3})}\\
&\leq \frac{c_2[\|\xi(t)\|^q+\|\xi(t)\|^p]}{\|(h_t)'(\xi(t))\|^2_*}
 2\|(h_t)'(\xi(t))\|_*\quad
\text{(by hypothesis and \eqref{eq2})}\\
&\leq \frac{2c_2}{\delta_0}(1+\|\xi(t)\|)\quad \text{(see \eqref{eq5})}
\end{align*}
It follows that the local flow $\xi(\cdot,u)$ is in fact global on
$[0,1]$ (see \cite[theorem 5.1.22, p.618]{GPap1}).

We go back in denoting the flow by $\xi(t,u)$. For every $t\in [0,1]$,
$\xi(t,u)$ is an homeomorphism. Hence $\xi(1,\cdot)$ is a homeomorphism of
$h_0^{\theta}$ onto a subset $D_0$ of $h_1^{\theta}$. Reversing the time
(that is replacing $t$ by $1-t$) and using the corresponding global flow
$\xi_{*}(t,v)$, we have that $h_1^{\theta}$ is a homeomorphic to a subset
$D_1$ of $h_0^{\theta}$. We set
$$
\eta(t,u)=\xi_{*}(t,\xi(t,u))\quad\text{for all }(t,u)\in [0,1]\times h_0^{\theta}.
$$
We have that
\begin{equation}\label{eq6}
\eta(0,\cdot)\text{ is homotopy equivalent to $id|_{D_0}(\cdot)$ and
$\eta(1,\cdot)=(\xi_{*})_1\circ \xi_1$}.
\end{equation}
In a similar fashion, if we set
$$
\eta_{*}(t,v)=\xi(t,\xi_{*}(t,v))\quad\text{for all }
(t,v)\in [0,1]\times h_1^{\theta},
$$
then we have that
\begin{equation}\label{eq7}
\eta_{*}(0,\cdot)\text{ is homotopy equivalent to $id|_{D_1}(\cdot)$ and
$\eta_*(1,\cdot)=\xi_1\circ(\xi_{*})_1$}.
\end{equation}
Recall that
\begin{equation}\label{eq8}
\{D_0,h_0^{\theta}\}\text{ and }\{D_1,h_1^{\theta}\}\text{ are homeomorphic pairs}.
\end{equation}
Then from \eqref{eq6}, \eqref{eq7}, \eqref{eq8} it follows that:
$h_0^{\theta}$ and $h_1^{\theta}$ are homotopy equivalent,
which implies that
\[
H_k(X,h_0^{\theta})=H_k(X,h_1^{\theta})\quad\text{for all }k\in \mathbb{N}_0
\]
(see Motreanu-Motreanu-Papageorgiou
\cite[Proposition 6.11, p.143]{MotreanuMotreanuPap}),
which implies
\[
C_k(h_0,\infty)=C_k(h_1,\infty)\quad\text{for all }k\in \mathbb{N}_0
\]
(choosing $\theta\in \mathbb{R}$ even smaller if necessary).
\end{proof}

Next let us recall some basic facts concerning the spectrum of the Robin
r-Laplacian. So let $\beta\in C^{0,\alpha}(\partial\Omega)$, $0<\alpha<1$,
$\beta(z)\geq 0$ for all $z\in \partial \Omega$ and consider the
nonlinear eigenvalue problem
\begin{equation}\label{eq9}
\begin{gathered}
-\Delta_ru(z)=\widehat{\lambda}|u(z)|^{r-2}u(z)\quad\text{in }\Omega, \\
\frac{\partial u}{\partial n_{r}}+\beta (z)|u|^{r-2}u=0 \quad\text{on }
\partial \Omega.
\end{gathered}
\end{equation}
This eigenvalue problem was studied by Papagerogiou-Radulescu \cite{PaRa4}.
A number $\widehat{\lambda}\in \mathbb{R}$ is an eigenvalue of the negative
Robin r-Laplacian, if problem \eqref{eq9} admits a nontrivial solution
$\widehat{u}$. The nontrivial solution $\widehat{u}$ is an eigenfunction
corresponding to the eigenvalue $\widehat{\lambda}$.
There is a smallest eigenvalue $\widehat{\lambda}_1(r)$ which has the
following properties:
\begin{itemize}
 \item $\widehat{\lambda}_1(r)\geq 0$ and it is isolated (in fact if $\beta=0$, 
then $\widehat{\lambda}_1(r)=0$, while if $\beta\neq 0$, then 
$\widehat{\lambda}_1(r)>0$)
 \item $\widehat{\lambda}_1(r)$ is simple (this means that, if 
$\widehat{u},\widehat{v}$ are eigenfunctions corresponding to 
$\widehat{\lambda}_1(r)$, then $\widehat{u}=c\widehat{v}$ for some 
$c\in \mathbb{R}\backslash\{0\}$).

 \item
 \begin{equation}\label{eq10}
 \widehat{\lambda}_1(r)=\inf\Big[\frac{\|Du\|_r^r+\int_{\partial \Omega}
\beta(z) |u|^r \,d\sigma}{\|u\|_r^r}:u\in W^{1,r}(\Omega),\;u\neq 0\Big].
 \end{equation}
 \end{itemize}

In \eqref{eq10} the infimum is attained on the corresponding one dimensional
eigenspace. From \eqref{eq10} it is easy to see that the elements of this
eigenspace have fixed sign. By $\widehat{u}_1(r)$ we denote the $L^{r}$-normalized
(that is, $\|\widehat{u}_1(r)\|_r=1$) positive eigenfunction corresponding
to $\widehat{\lambda}_1(r)$. The nonlinear regularity theory and the nonlinear
maximum principle (see for example, Gasinski-Papageorgiou \cite[pp.737-738]{GPap1})
imply that $\widehat{u}_1(r)\in D_+$. Since the spectrum $\widehat{\sigma}(r)$
of \eqref{eq9} is closed and $\widehat{\lambda}_1(r)$ is isolated, then the
second eigenvalue $\widehat{\lambda}_2(r)$ is well defined by
$$
\widehat{\lambda}_2(r)=\inf[\widehat{\lambda}
\in \widehat{\sigma}(r):\widehat{\lambda}>\widehat{\lambda}_1(r)].
$$
By $\operatorname{ind}(\cdot)$ we denote the $\mathbb{Z}_2$-cohomological
index of Fadell-Rabinowitz (see \cite{CingolaniDegiovanni}).
Using $\operatorname{ind}(\cdot)$ and the Ljusternik-Schnirelmann minimax
scheme, we can define a whole sequence
$\{\widehat{\lambda}_k(r)\}_{k\in \mathbb{N}}$ of dinstict eigenvalues
of \eqref{eq9}, by setting
$$
\widehat{\lambda}_k(r)=\inf[\sup_{u\in A}\{\|Du\|_r^r
+\int_{\partial \Omega}\beta(z)|u|^r \,d\sigma\}:A\subseteq M\;\text{symmetric,}
\operatorname{ind}(A)\geq k,k\in \mathbb{N}]
$$
with $M=\{u\in W^{1,r}(\Omega):\|u\|_r=1\}$. Evidently M is a $C^1$-Banach manifold.
We have $\widehat{\lambda}_k(r)\to +\infty$ as $k\to +\infty$ and these
eigenvalues are known as ``variational eigenvalues'' of \eqref{eq9}.
If $k=1,2$, then $\widehat{\lambda}_k(r)$ are as defined earlier. We do not
know if this sequence exhausts $\widehat{\sigma}(r)$. This is the case if
$r=2$ (linear eigenvalue problem) and if $N=1$ (ordinary differential equations).
We mention that, if $\widehat{u}$ is an eigenfunction corresponding to the
eigenvalue $\widehat{\lambda}_k(r)$, $k\geq 2$, then
$\widehat{u}\in C^1(\overline{\Omega})$ (by the nonlinear regularity theory)
and it is nodal (that is, sign-changing),

Also, for the nonprincipal eigenvalues, the corresponding eigenspaces are only
cones and not linear subspaces of $W^{1,r}(\Omega)$ and the latter cannot
be expressed as a direct sum of these eigenspaces. For these reasons,
when $r\neq 2$ (nonlinear eigenvalue problem), it is difficult to deal with
problems resonant with respect to any nonprincipal eigenvalue.

As an easy consequence of the properties of the principal eigenvalue
$\widehat{\lambda}_1(r)$, we have the following result
(see Mugnai-Papageorgiou \cite[ Lemma 4.11]{MugnaiPap1}).

\begin{lemma}\label{lem5}
If $\theta \in L^{\infty}(\Omega)$, $\theta (z)\leq \widehat{\lambda}_1(r)$
for a.a. $z\in \Omega$ and $\theta \neq \widehat{\lambda}_1(r)$,
then there exists $\widehat{c}>0$ such that
$$
\|Du\|_r^r+\int_{\partial \Omega}\beta(z)|u|^r \,d\sigma
-\int_{\Omega}\theta(z)|u|^r\,dz\geq \widehat{c}\|u\|^r
$$
for all $u\in W^{1,p}(\Omega)$.
\end{lemma}

For $x\in \mathbb{R}$, let $x^{\pm}=\max\{\pm x,0\}$.
Then for $u\in W^{1,r}(\Omega)$ we set $u^{\pm}(\cdot)=u(\cdot)^{\pm}$.

We know that
$$
u^{\pm}\in W^{1,r}(\Omega), \;u=u^{+}-u^{-},\;|u|=u^{+}+u^{-}
$$
Let us introduce our hypotheses on the data of \eqref{eq1}.
\begin{itemize}
 \item[(H1)] $f:\Omega\times\mathbb{R}\to
 \mathbb{R}$ is a Caratheodory function such that $f(z,0)=0$, for a.a.
$z\in \Omega$ and
 \begin{itemize}
 \item[(i)] for every $\rho>0$, there exists $\alpha_{\rho}\in L^{\infty}(\Omega)_+$
such that
 $$
|f(z,x)|\leq \alpha_{\rho}(z)\quad \text{for a.a. $z\in \Omega$ all }|x|\leq \rho;
$$
 \item[(ii)] there exists an integer $m\geq 2$ such that
 $$
\lim_{x\to \pm\infty}\frac{f(z,x)}{|x|^{p-2}x}
=\widehat{\lambda}_m(p)\;\;\text{uniformly for a.a. }\;\;z\in \Omega;
$$
 \item[(iii)] if $F(z,x)=\int_0^xf(z,s)ds$, then
\[
0<c_0\leq\liminf_{x\to \pm\infty}\frac{f(z,x)x-pF(z,x)}{|x|^{p-1}}
\quad \text{uniformly for a.a. } z\in \Omega;
\]

\item[(iv)] there exists a function $\theta \in L^{\infty}(\Omega)_+$ such that
$\theta (z)\leq \widehat{\lambda}_1(q)$ for a.a. $z\in \Omega$, with strict
inequality on a set of positive measure,
\[
\limsup_{x\to 0}\frac{qF(z,x)}{|x|^{q}}\leq \theta(z)\quad
\text{uniformly for a.a. } z\in \Omega
\]
note that if $\beta_2\equiv 0$, then $\widehat{\lambda}_1(q)=0$).
\end{itemize}
 \end{itemize}

\begin{remark} \rm
Hypothesis (H1)(ii) implies that at $\pm\infty$ we have resonance with respect
to a nonprincipal variational eigenvalue. We can write that
\begin{equation}\label{eq11}
f(z,x)=\widehat{\lambda}_m(p)|x|^{p-2}x+f_0(z,x)
\end{equation}
with a Caratheodory function $f_0(z,x)$ such that
\begin{equation}\label{eq12}
\lim_{x\to \pm\infty}\frac{f_0(z,x)}{|x|^{p-2}x}=0\text{ uniformly for a.a. }
z\in \Omega.
\end{equation}
If we set $F_0(z,x)=\int_0^x f_0(z,s) ds$. Then
$$
F(z,x)=\frac{\widehat{\lambda}_m(p)}{p}|x|^p+F_0(z,x)
$$
and we have
\begin{equation}\label{eq13}
0<c_0\leq\liminf_{x\to \pm\infty}\frac{f_0(z,x)x-pF_0(z,x)}{|x|^{p-1}}
\quad \text{uniformly for a.a. } z\in \Omega;
\end{equation}
see hypothesis (H1)(iii).
\end{remark}

\begin{example} \label{examp1} \rm
The following function satisfies hypotheses (H1).
For the sake of simplicity we drop the $z$-dependence
$$
f(x)=  \begin{cases}
\theta|x|^{q-2}x & \text{if } |x|\leq 1 \\
\widehat{\lambda}_m(p)|x|^{p-2}x+|x|^{\tau-2}x+\widehat{c}
& \text{if } 1<|x|
\end{cases}
$$
with $0<\theta<\widehat{\lambda}_1(q)$,
$\widehat{c}=\theta-(\widehat{\lambda}_m(p)+1)$, $1<q<p$, $p-1\leq \tau<p$.
\end{example}

The hypotheses on the boundary coefficient $\beta(\cdot)$ are the following
 \begin{itemize}
 \item[(H2)] $\beta_1\in C^{0,\alpha}(\partial \Omega)$,
$\beta_2\in C^{0,\eta}(\partial \Omega)$ with
$0<\alpha,\eta<1$ and $\beta_1(z),\beta_2(z)\geq 0$ for all $z\in\partial\Omega$.

 \item[(H3)] For every $\rho>0$ there exists $\widehat{\xi}_{\rho}>0$ such that
 $$
f(z,x)x+\widehat{\xi}_{\rho}(|x|^{\rho}+|x|^q)\geq 0\quad\text{for a.a. }
x\in \Omega,\;\;|x|\leq \rho.
$$
\end{itemize}

\begin{remark} \rm
Note that when $\beta_1=\beta_2=0$ in (H2), we recover the Neumann problem.
Also note that the example given earlier satisfies (H3).
\end{remark}

In what follows for $\tau\in (1,+\infty)$, we introduce the $C^1$-functional
$\gamma_{\tau}:W^{1,\tau}(\Omega)\to \mathbb{R}$ defined by
$$
\gamma_{\tau}(u)=\|Du\|_{\tau}^{\tau}+\int_{\partial \Omega}
\beta(z)|u|^{\tau}\,d\sigma\quad\text{for all }u\in W^{1,\tau}(\Omega).
$$
Let $\varphi:W^{1,p}(\Omega)\to \mathbb{R}$ be the energy functional for
problem \eqref{eq1} defined by
$$
\varphi(u)=\frac{1}{p}\gamma_p(u)+\frac{1}{q}\gamma_q (u)
-\int_{\Omega} F(z,u)dz\;\;\text{for all}\;u\in W^{1,p}(\Omega),
$$
$\gamma_p$ with $\beta=\beta_1$ and $\gamma_q$ with $\beta=\beta_2$.
Evidently $\varphi\in C^1(W^{1,p}(\Omega))$.

\section{Solutions of constant sign}

We introduce the following truncations-perturbations of the reaction term
$f(z,\cdot)$:
\begin{equation}\label{eq14}
\begin{gathered}
\widehat{f}_+(z,x)= \begin{cases}
 0 & \text{if } x\leq 0 \\
 f(z,x)+x^{p-1} & \text{if } 0<x ;
\end{cases} \\
\widehat{f}_{-}(z,x)=  \begin{cases}
f(z,x)+|x|^{p-2}x & \text{if } x< 0 \\
0 & \text{if } 0\leq x
\end{cases}
\end{gathered}
\end{equation}
Both are Caratheodory functions. We set
$$
\widehat{F}_+(z,x)=\int_0^x \widehat{f}_+(z,s)ds\quad\text{and}\quad
\widehat{F}_{-}(z,x)=\int_0^x \widehat{f}_{-}(z,s)ds
$$
and consider the $C^1$-functionals
$\widehat{\varphi}_{\pm}:W^{1,p}(\Omega)\to \mathbb{R}$ defined by
$$
\widehat{\varphi}_{\pm}(u)=\frac{1}{p}\gamma_p (u)+\frac{1}{q}\gamma_q
(u)+\frac{1}{p}\|u\|_p^p-\int_{\Omega}\widehat{F}_{\pm}(z,u)dz
\quad\text{for all }u\in W^{1,p}(\Omega.)
$$

\begin{proposition}\label{prop6}
If hypotheses {\rm(H1)(i)--(iii), (H2)} hold then the functionals
$\widehat{\varphi}_{\pm}$ satisfy the C-condition.
\end{proposition}

\begin{proof}
We do the proof for the functional $\widehat{\varphi}_+$ the proof for
$\widehat{\varphi}_{-}$ being similar.
So, let $\{u_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)$ be a sequence such that
\begin{gather}\label{eq15}
|\widehat{\varphi}_+(u_n)|\leq M_1\quad \text{for some }M_1>0,\text{ and all }
n\in \mathbb{N}, \\
\label{eq16}
(1+\|u_n\|)\widehat{\varphi}'_{+}(u_n)\to 0 \quad \text{in }
W^{1,p}(\Omega)^{*}\quad \text{as }n\to \infty.
\end{gather}
From \eqref{eq16} we have
\begin{equation} \label{eq17}
\begin{aligned}
&\Big|\langle A_{p}(u_n),h\rangle
 +\langle A_{q}(u_n),h\rangle+\int_{\partial \Omega} (\beta_1(z)|u_n|^{p-2}u_n
 +\beta_2 (z) |u_n|^{q-2}u_n) h \,d\sigma \\
&+\int_{\Omega}|u_n|^{p-2}u_n h\,dz-\int_{\Omega}\widehat{f}_+(z,u_n)h\,dz\Big|\\
&\leq \frac{\varepsilon_n\|h\|}{1+\|u_n\|}\quad
\text{for all }h\in W^{1,p}(\Omega)\text{ with }\varepsilon_n\to 0^+.
\end{aligned}
\end{equation}

In \eqref{eq17} we choose $h=-u_n^{-}\in W^{1,p}(\Omega)$. Then
\[
\|Du_n^{-}\|_p^p +\|Du_n^{-}\|_q^q + \|u_n^{-}\|_p^p
\leq \varepsilon_n\quad\text{for all }n\in \mathbb{N}
\]
(see \eqref{eq14} and hypothesis (H2)) which implies
\begin{equation}\label{eq18}
 u_n^{-}\to 0\quad\text{in } W^{1,p}(\Omega) \text{ as } n\to \infty.
\end{equation}
\smallskip

\noindent\textbf{Claim 1:}
$\{u_n^{+}\}_{n\in \mathbb{N}}\subseteq W^{1,p}(\Omega)$ is bounded.

We argue indirectly. So, suppose that Claim 1 is not true.
Then by passing to a subsequence if necessary, we may assume that
$\|u_n^{+}\|\to \infty$. We set $y_n=\frac{u_n^{+}}{\|u_n^{+}\|}\;n\in \mathbb{N}$.
Then
$\|y_n\|=1$, $y_n\geq 0$ for all $n\in \mathbb{N}$ and so we may assume that
\begin{equation}\label{eq19}
y_n\stackrel{w} \to y\text{ in } W^{1,p}(\Omega)\quad\text{and}\quad
y_n\to y\text{ in }L^p(\Omega)\text{ and in }L^p(\partial \Omega).
\end{equation}

From \eqref{eq17} and \eqref{eq18}, we have
\begin{align*}
&\Big|\langle A_{p}(u_n^{+}),h\rangle +\langle A_{q}(u_n^{+}),h\rangle
 +\int_{\partial \Omega} \Big(\beta_1(z)(u_n^{+})^{p-1}
 +\beta_2 (z)(u_n^{+})^{q-1}\Big) h \,d\sigma \\
&+\int_{\Omega}(u_n^{+})^{p-1} h\,dz-\int_{\Omega}\widehat{f}_+(z,u_n^{+}) h\,dz\Big|\\
& \leq {\varepsilon'_n}\|h\|\quad\text{for all }n\in\mathbb{N,}\text{ with }
{\varepsilon'_n}\to 0^{+},
\end{align*}
which implies
\begin{equation} \label{eq20}
\begin{aligned}
&\Big|\langle A_p (y_n)+\frac{1}{\|u_n^{+}\|^{p-q}}A_{q}(y_n),h\rangle
 +\int_{\partial \Omega}\Big(\beta_1(z)y_n^{p-1}+\frac{1}{\|u_n^{+}\|^{p-q}}
 \beta_2(z) y_n^{q-1}\Big) h \,d\sigma \\
&+\int_{\Omega}y_n^{p-1} h\,dz-\int_{\Omega}
 \frac{\widehat{f}_+(z,u_n^{+})}{\|u_n^{+}\|^{p-1}} h\,dz\Big| \\
&\leq {\varepsilon'_n}\frac{\|h\|}{\|u_n^{+}\|^{p-1}}\quad
\text{for all }n\in \mathbb{N}.
\end{aligned}
\end{equation}
In \eqref{eq20} we choose $h=y_n-y\in W^{1,p}(\Omega)$, pass to the limit as
$n\to \infty$ and use \eqref{eq19} and the fact that $q<p$. Then
\[
\lim_{n\to \infty}\langle A_p(y_n),y_n-y\rangle =0
\]
which implies
\begin{equation} \label{eq21}
 y_n\to y\quad \text{in }W^{1,p}(\Omega) \text{ as } n\to  \infty,
\end{equation}
\ thus $\|y\|=1$ and $y\geq 0$, see Proposition \ref{prop2}.

Hypotheses (H1)(i),(ii) imply that
\[
|f(z,x)|\leq c_3(1+|x|^{p-1})\quad \text{for a.a. $z\in \Omega$, all
$x\in \mathbb{R}$ and some $c_3>0$},
\]
which in turn implies
\[
\Big\{\frac{\widehat{f}_{+}(\cdot,u_n^{+}(\cdot))}{\|u_n^{+}\|^{p-1}}
\Big\}_{n\in \mathbb{N}}\subseteq L^{p'}(\Omega)\text{ is bounded (see \eqref{eq14})}.
\]
Therefore using hypothesis (H1)(ii) we have (at least for a subsequence)
such that
\begin{equation}\label{eq22}
\frac{\widehat{f}_{+}(\cdot,u_n^{+}(\cdot))}{\|u_n^{+}\|^{p-1}}
\stackrel{w} \to (\widehat{\lambda}_m(p)+1)y^{p-1}\quad\text{in }
L^{p'}(\Omega)\text{ as } n\to \infty
\end{equation}
(see Aizicovici-Papageorgiou \cite[proof of Proposition 30]{AizicPapStaicu1}).

If in \eqref{eq20} we pass to the limit as $n\to \infty$ and use \eqref{eq21}
and \eqref{eq22} and the facts that $q<p$, $y\geq 0$, then
\[
\langle A_p(y),h\rangle +\int_{\partial \Omega}\beta_1(z) y^{p-1} h \,d\sigma
=\widehat{\lambda}_m(p)\int_{\Omega} y^{p-1} h\,dz\quad
\text{for all $h\in W^{1,p}(\Omega)$ (see \eqref{eq14})}
\]
which implies
\begin{equation} \label{eq23}
\begin{gathered}
 -\Delta_p y(z)=\widehat{\lambda}_m(p)|y(z)|^{p-2} y(z) \quad \text{for a.a. }
z\in \Omega, \\
\frac{\partial y}{\partial \eta_{p}}+\beta(z) y^{p-1}=0\quad \text{on }
\partial \Omega
\end{gathered}
\end{equation}
(see Papageorgiou-Radulescu \cite{PaRa4}).
Since $m\geq 2$, from \eqref{eq23} we infer that $y$ must be nodal,
 which contradicts \eqref{eq21}. This proves Claim 1.

From \eqref{eq18} and  Claim 1, we have that
$\{u_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)$ is bounded. So, we may assume that
\begin{equation}\label{eq24}
u_n\stackrel{w} \to u\text{ in }W^{1,p}(\Omega)\quad
\text{and}\quad u_n\to u\text{ in $L^{p}(\Omega)$ and in }
L^{p}(\partial\Omega).
\end{equation}

In \eqref{eq17} we choose $h=u_n-u\in W^{1,p}(\Omega)$, pass to the limit
as $n\to \infty$ and use \eqref{eq24}. Then we have
\begin{gather*}
\lim_{n\to \infty}[\langle A_p (u_n), u_n-u \rangle
+\langle A_{q}(u_n),u_n-u\rangle ]=0,\\
\Rightarrow \limsup_{n\to \infty}[\langle A_p (u_n), u_n-u \rangle
+\langle A_{q}(u),u_n-u\rangle ]\leq 0,
\end{gather*}
(recall that $A_{q}(\cdot)$ is monotone),
which implies
\begin{gather*}
 \limsup_{n\to \infty}[\langle A_p (u_n), u_n-u \rangle
\leq 0\quad \text{(see \eqref{eq24}),}\\
\Rightarrow u_n\to u\;\;\text{in}\;\;W^{1,p}(\Omega)\quad
\text{(see Proposition \ref{prop2})}.
\end{gather*}
This proves that the functional $\widehat{\varphi}_{+}$ satisfies the
C-condition. Similarly for the functional $\widehat{\varphi}_{-}$.
\end{proof}

\begin{proposition}\label{prop7}
If hypotheses {(H1)(i)--(iii), (H2)} hold, then the energy functional
$\varphi$ satisfies the C-condition.
\end{proposition}

\begin{proof}
Let $\{u_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)$ be a sequence such that
\begin{gather}\label{eq25}
|\varphi(u_n)|\leq M_2\quad \text{for some } M_2>0,\text{and  all }n\in \mathbb{N},\\
\label{eq26}
(1+\|u_n\|)\varphi'(u_n)\to 0\quad \text{in }W^{1,p}(\Omega)^*\text{ as }
n\to \infty.
\end{gather}
From \eqref{eq26} we have
\begin{equation}\label{eq27}
\begin{aligned}
&\Big|\langle A_p (u_n),h\rangle+\langle A_{q}(u_n),h\rangle
+\int_{\partial\Omega} (\beta_1 (z)|u_n|^{p-2} u_n
 +\beta_2(z)|u_n|^{q-2} u_n) h \,d\sigma \\
&-\int_{\Omega}f(z,u_n) hdz\Big| \\
&\leq \frac{\varepsilon_n \|h\|}{1+\|u_n\|}\quad
\text{for all } h\in W^{1,p}(\Omega),\text{ with }\varepsilon_n\to 0^+.
\end{aligned}
\end{equation}
In \eqref{eq27} we choose $h=u_n\in W^{1,p}(\Omega)$. Then
\begin{equation}\label{eq28}
-\gamma_p(u_n)-\gamma_q(u_n)+\int_{\Omega}f(z,u_n) u_n\,dz
\leq \varepsilon_n\quad\text{for all }n\in \mathbb{N}.
\end{equation}
Also from \eqref{eq25} we have
\begin{equation}\label{eq29}
\gamma_p(u_n)+\frac{p}{q}\gamma_q(u_n)-\int_{\Omega}p F(z,u_n)\,dz
\leq p M_2\quad\text{for all }n\in \mathbb{N}.
\end{equation}
Adding \eqref{eq28} and \eqref{eq29}, we obtain
\begin{equation}\label{eq30}
\big(\frac{p}{q}-1\big) \gamma_q (u_n)
+\int_{\Omega}[f(z,u_n)u_n-pF(z,u_n)]\,dz\leq M_3
\end{equation}
for some $M_3>0$ and all $n\in \mathbb{N}$,
\[
 \int_{\Omega} [f(z,u_n) u_n - p F(z,u_n)]\leq M_3\quad
\text{for all }n\in \mathbb{N},
\]
(recall that $q<p$ and $\gamma_{q}\geq 0$).
\smallskip

\noindent\textbf{Claim 2:}
 $\{u_n\}_n\geq 1\subseteq W^{1,p}(\Omega)$ is bounded,

We argue by contradiction. So, suppose that  Claim 2 is not true.
We may assume that $\|u_n\|\to \infty$. We set
$y_n=\frac{u_n}{\|u_n\|}$ $n\in \mathbb{N}$. Then $\|y_n\|=1$ for all
$n\in \mathbb{N}$ and so we may assume that
\begin{equation}\label{eq31}
y_n\stackrel{w} \to y\text{ in }W^{1,p}(\Omega)\quad\text{and}\quad
y_n\to y\text{ in } L^p(\Omega)\text{ and in }L^{p}(\partial \Omega).
\end{equation}

From \eqref{eq27} we have
\begin{equation} \label{eq32}
\begin{aligned}
&\Big|\langle A_p(y_n)+\frac{1}{\|u_n\|^{p-2}}  A_{q}(y_n),h\rangle\\
&+\int_{\partial \Omega}\Big(\beta_1 (z)|y_n|^{p-2}y_n
 +\frac{\beta_2(z)}{\|u_n\|^{p-q}}|y_n|^{q-2}y_n \Big)h\,dz 
-\int_{\Omega}\frac{f(z,u_n)}{\|u_n\|^{p-1}} h\,dz\Big| \\
&\leq \frac{\varepsilon_n \|h\|}{(1+\|u_n\|)\|u_n\|^{p-1}}\quad\text{for all }
n\in \mathbb{N}.
\end{aligned}
\end{equation}
In \eqref{eq32} we choose $h=y_n-y\in W^{1,p}(\Omega)$, we pass to the
limit as $n\to \infty$ and use the fact that $q<p$. Then
$\lim_{n\to \infty}\langle A_p (y_n),y_n-y\rangle =0$ which implies
\begin{equation}\label{eq33}
 y_n\to y \quad\text{in }W^{1,p}(\Omega)\text{ and so }\|y\|=1
\quad \text{(see Proposition \ref{prop2})}
\end{equation}
From \eqref{eq33} it follows that we can find $D\subseteq \Omega$
measurable with $|D|_{N}>0$ (here $|\cdot|_{N}$ denotes the Lebesgue
measure on $\mathbb{R}^N$) such that
\begin{equation}\label{eq34}
|u_n(z)|\to +\infty\quad\text{for all }z\in D.
\end{equation}
Hypotheses (H1)(i), (iii) imply that we can find $c_4>0$ such that
\begin{equation}\label{eq35}
-c_4\leq f(z,x) x-pF(z,x)\quad\text{for a.a. } z\in \Omega,\text{ all }\mathbb{R}.
\end{equation}
Then
\begin{equation} \label{eq36}
\begin{aligned}
&\int_{\Omega} [f(z,u_n)u_n-p F(z,u_n)]dz \\
&=\int_{\Omega\backslash{D}} [f(z,u_n) u_n-p F(z,u_n)]dz
 +\int_{D} [f(z,u_n) u_n - p F(z,u_n)]\,dz \\
&\geq -c_4 |\Omega\backslash {D}|_{N}+\int_{D}[f(z,u_n)u_n-pF(z,u_n)]\,dz
\end{aligned}
\end{equation}
for all $n\in \mathbb{N}$ (see \eqref{eq35}).
From \eqref{eq34}, hypothesis (H1)(iii) and Fatou's lemma, we have
\[
\int_{D} [f(z,u_n) u_n -p F(z,u_n)]\,dz\to +\infty,
\]
which implies
\begin{equation} \label{eq37}
 \int_{\Omega} [f(z,u_n) u_n - p F(z,u_n)]\,dz\to +\infty\quad
\text{(see \eqref{eq36})}.
\end{equation}
Comparing \eqref{eq30} and \eqref{eq37} we have a contradiction.
This proves the claim 2.

Using the claim 2, we may assume that
\begin{equation}\label{eq38}
u_n\stackrel{w} \to u\text{ in }W^{1,p}(\Omega) \quad
\text{and}\quad
u_n\to u\text{ in }L^{p}(\Omega)\text{ and in } L^{p}(\partial \Omega).
\end{equation}
In \eqref{eq27} we choose $h=u_n-u\in W^{1,p}(\Omega)$, we pass to the limit
as $n\to \infty$ and use \eqref{eq38}. Then
\begin{gather*}
 \lim_{n\to \infty}[\langle A_{p}(u_n),u_n-u\rangle+\langle A_{q}(u_n),u_n-u\rangle ]
=0,\\
\Rightarrow   \limsup_{n\to \infty} \langle A_{p}(u_n),u_n-u\rangle \leq 0,\quad
\text{(using the monotonicity of $A_{q}(\cdot)$ and \eqref{eq38})},\\
\Rightarrow u_n\to u\quad\text{in }\quad W^{1,p}(\Omega)\quad
\text{(see Proposition \ref{prop2})},
\end{gather*}
Therefore  $\varphi$ satisfies the C-condition.
\end{proof}

Hypotheses (H1)(i), (ii) imply that
\begin{equation}\label{eq39}
|f(z,x)|\leq c_5 (1+|x|^{p-1})\quad \text{for a.a. }
z\in \Omega\text{ all }x\in \mathbb{R},\text{ some }c_5>0\,.
\end{equation}

\begin{proposition}\label{prop8}
If hypotheses {\rm (H1)(iv)}, \eqref{eq39} and {\rm (H2)} hold, then $u=0$ is a
local minimizer of the functionals $\widehat{\varphi}_{\pm}$ and of $\varphi$.
\end{proposition}

\begin{proof}
We do the proof for the functional $\widehat{\varphi}_+$, the proofs for
$\widehat{\varphi}_{-}$ and $\varphi$ being similar.
Hypothesis (H1)(iv) implies that given $\varepsilon>0$, we can find
$\delta=\delta(\varepsilon)>0$ such that
\begin{equation}\label{eq40}
F(z,x)\leq \frac{1}{q}[\theta(z)+\varepsilon]|x|^{q}\quad\text{for a.a. }
z\in \Omega,\text{ all }|x|\leq \delta.
\end{equation}
On the other hand, given $r>p>q$, using \eqref{eq39} we see that we can find
$c_6=c_6(\varepsilon)>0$ such that
\begin{equation}\label{eq41}
F(z,x)\leq c_6|x|^{r}\quad \text{for a.a. }z\in \Omega,\text{ all }
|x|\geq \delta.
\end{equation}
Since $\theta\in L^{\infty}(\Omega)_+$, from \eqref{eq40} and \eqref{eq41}
it follows that
\begin{equation}\label{eq42}
F(z,x)\leq \frac{1}{q}[\theta(z)+\varepsilon]|x|^{q}
+c_6 |x|^{r}\quad \text{for a.a. }z\in \Omega,\text{ all }x\in \mathbb{R}.
\end{equation}
For $u\in W^{1,p}(\Omega)$ we have
\begin{align*}
\widehat{\varphi}_+(u)
&=\frac{1}{p}\gamma_{p}(u)+\frac{1}{q}\gamma_{q}(u)+\frac{1}{p}\|u^{-}\|_p^p
 -\int_{\Omega} F(z,u^{+})\,\,dz\quad \text{(see \eqref{eq14})}\\
&\geq \frac{1}{p}\gamma_{p}(u^{+})+\frac{1}{p}
 \big[\gamma_p (u^{-})+\|u^{-}\|_p^p\big]\\
&\quad +\frac{1}{q}\Big[\gamma_{q}(u^{+})
 -\int_{\Omega}\theta(z)(u^{+})^{q}\,dz-\varepsilon \|u^{+}\|^q\Big]
-c_7 \|u\|^{r}
\end{align*}
for some $c_7>0$ (see \eqref{eq42}).
Using Lemma \ref{lem5} and choosing $\varepsilon >0$ small, we have
\[
\widehat{\varphi}_+(u)\geq \frac{1}{p}\|u^{-}\|^p
+\frac{c_8}{q}\|u^{+}\|^q-c_7\|u\|^r\]
for some $c_8>0$.
If $\|u\|\leq 1$, then $\|u^{+}\|$, $\|u^{-}\|\leq 1$ and so
$\|u^{+}\|^{q}\geq \|u^{+}\|^p$. Hence
\begin{equation} \label{eq44}
\begin{aligned}
\widehat{\varphi}_+(u)
&\geq\frac{1}{p}\|u^{-}\|^p+\frac{c_8}{q}\|u^+\|^p-c_7\|u\|^{r} \\
&\geq c_9\|u\|^p-c_7\|u\|^r\quad \text{for some }c_9>0.
\end{aligned}
\end{equation}
Since $r>p$, if we choose $\rho\in (0,1)$ small, then from \eqref{eq44}
 we see that
\[
\widehat{\varphi}_+(u)\geq 0\quad\text{for all }u\in W^{1,p}(\Omega)
\text{ with }\|u\|\leq \rho,
\]
which implies that $u=0$ is a local minimizer of
$\widehat{\varphi}$.
Similar argument works for the functionals $\widehat{\varphi}_{-}$ and $\varphi$.
\end{proof}

\begin{remark} \label{rmk4} \rm
We can avoid the use of \eqref{eq39} and instead assume that
$$
|f(z,x)|\leq c_{10}(1+|x|^{p^*-1})\quad\text{for a.a. }z\in \Omega,\text{ all }
x\in\mathbb{R},\text{ some }c_{10}>0.
$$
For $u\in C^1(\overline{\Omega})$ with $\|u\|_{C^1(\overline{\Omega})}\leq \delta$,
we have
$$
\widehat{\varphi}_+(u)\geq \frac{1}{p}\|u^{-}\|^p
+\frac{c_8}{q}\|u^+\|^q\quad \text{(see \eqref{eq40} and Lemma \ref{lem5})}
$$
By taking $\delta>0$ even smaller, we have
\[
\widehat{\varphi}_+(u)\geq \frac{1}{p}\|u^{-}\|^p+c_{11}\|u^+\|^p
\geq c_{12}\|u\|^p
\]
which implies that $u=0$ is a local $C^1(\overline{\Omega})$-minimizer of
$\widehat{\varphi}_+$, and that
$u=0$ is a local $W^{1,p}(\Omega)$-minimizer of $\widehat{\varphi}_+$
(see Proposition \ref{prop4}).
\end{remark}

Now we are ready to produce two constant sign solutions
(one positive and the other negative).

\begin{proposition}\label{prop9}
If hypotheses {\rm(H1)--(H3)} hold, then problem \eqref{eq1} has at least
two nontrivial constant sign smooth solutions
$$
u_0\in D_+\quad \text{and}\quad v_0\in -D_{+}.
$$
\end{proposition}

\begin{proof}
Using \eqref{eq14} and the nonlinear regularity theory of Lieberman
\cite{aLieberman1991}, we have
\begin{equation}\label{eq45}
K_{\widehat{\varphi}_+}\subseteq C_+\quad\text{and}\quad
K_{\widehat{\varphi}_{-}}\subseteq -C_{+}.
\end{equation}
So, we assume that both sets are finite or otherwise we already have two
 sequences consisting of distinct positive and negative solutions which in
fact belong in $D_+$ and $D_{-}$ respectively (see the last part of this proof).

First, we prove the existence of a positive solution.
Since $K_{\widehat{\varphi}_+}$ is finite and $u=0$ is a local minimizer
of $\widehat{\varphi}_+$ (see Proposition \ref{prop8}), we can find
$\rho\in (0,1)$ small such that
\begin{equation}\label{eq46}
\widehat{\varphi}_+(0)=0<\inf[\widehat{\varphi}_+(u):\|u\|=\rho]
=\widehat{m}^{+}_{\rho}
\end{equation}
(see Aizicovici-Papageorgiou-Staicu \cite[proof of Proposition 29]{AizicPapStaicu1}).
 Also because $m\geq 2$ and $q<p$, we see that
\begin{equation}\label{eq47}
\widehat{\varphi}_+(t \widehat{u}_1(p))\to -\infty\quad\text{as }t\to +\infty.
\end{equation}

Then \eqref{eq46}, \eqref{eq47} and Proposition \ref{prop6}, permit the use of
Theorem \ref{thm1} (the mountain pass theorem). So, we can find
$u_0\in W^{1,p}(\Omega)$ such that
\[
u_0\in K_{\widehat{\varphi}_+}\subseteq C_+
\text{ (see \eqref{eq45})}\quad\text{and}\quad
\widehat{m}^{+}_{\rho}\leq \widehat{\varphi}_+(u_0),
\]
which implies $u_0\in C_+\backslash\{0\}$ (see \eqref{eq46}).

We have
\begin{equation}\label{eq48}
\begin{gathered}
-\Delta_pu_0(z)\,-\Delta_q u_0(z)=f(z,u_0(z))\quad\text{for a.a. }z\in \Omega, \\
\frac{\partial u_0}{\partial n_{pq}}+\beta_1 (z)u_0^{p-1}+\beta_2(z)u_0^{q-1}=0
\quad\text{on }\partial \Omega.
\end{gathered}
\end{equation}
(see Papageorgiou-Radulescu \cite{PaRa4}).
Hypotheses (H1)(i),(iv) and (H3) imply that given $\rho>0$, we can find
$\widehat{\xi}_{\rho}>0$ such that
$$
f(z,x) x+\widehat{\xi}_{\rho}[|x|^p+|x|^q]\geq 0\quad\text{for a.a. }
z\in \Omega,\text{ all }|x|\leq \rho
$$
Let $\rho=\|u_0\|_{\infty}$. Then from \eqref{eq48} we have
\[
\Delta_p u_0(z)+\Delta_q u_0(z)\leq \widehat{\xi}_{\rho}
[ u_0(z)^{p-1}+u_0(z)^{q-1}]\quad\text{for a.a. }z\in \Omega,
\]
which implies $u_0\in D_+$ (see Pucci-Serrin
\cite[Theorem 5.4.1, p.111 and Theorem 5.5.1, p.120]{PuSe}).
Similarly, working this time with $\widehat{\varphi}_{-}$, we produce a
negative solution $v_0\in -D_+$.
\end{proof}

\section{Three solutions theorem}

In this section, using Morse Theory (critical groups), we produce a third nontrivial
smooth solution. For this purpose, using Proposition \ref{prop4},
we compute the critical groups of $\varphi$ and of $\widehat{\varphi}_{\pm}$
at infinity.

\begin{proposition}\label{prop10}
If hypotheses {\rm (H1)(i),(ii),(iii), (H2)} hold and
$K_{\varphi}$ is finite, then $C_m(\varphi,\infty)\neq 0$.
\end{proposition}

\begin{proof}
Let $\lambda\in (\widehat{\lambda}_m(p),\widehat{\lambda}_{m+1}(p)),
\lambda\notin \widehat{\sigma}(p)$ and consider the $C^1$-functional
$\psi:W^{1,p}(\Omega)\to \mathbb{R}$ defined by
$$
\psi(u)=\frac{1}{p}\gamma_p(u)-\frac{\lambda}{p}\|u\|_p^p\quad\text{for all }
u\in W^{1,p}(\Omega).
$$
We consider the homotopy $h_t(u)$ defined by
$$
h_t(u)=(1-t)\varphi(u)+t \psi(u)\quad\text{for all }
t\in (0,1),\;\text{all}\;u\in W^{1,p}(\Omega).
$$
Note that $h_0(\cdot)=\varphi(\cdot)$ and by Proposition \ref{prop7} $\varphi$
satisfies the C-condition. Also $h_1(\cdot)=\psi(\cdot)$ and since
$\lambda\notin \widehat{\sigma}(p)$, we see that $\psi$ satisfies the C-condition.
\smallskip

\noindent\textbf{Claim 3:}
 There exist $\theta_0\in \mathbb{R}$ and $\delta_0>0$ such that
$$
h_t(u)\leq \theta_0\Rightarrow (1+\|u\|)\|(h_t)'(u)\|_{*}
\geq \delta_0(\|u\|^{q}+\|u\|^{p})\quad\text{for all }t\in (0,1).
$$

As before we argue by contradiction. Since $(t,u)\to h_t(u)$ maps bounded
sets to bounded sets, if  Claim 3 is not true, we can find
$\{t_n\}_{n\geq 1}\subseteq [0,1]$ and $\{u_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)$
such that
\begin{equation}\label{eq49}
\begin{gathered}
t_n\to t,\;\|u_n\|\to +\infty,\; h_{t_n}(u_n)\to -\infty,\text{ and}\\
\|(h_{t_n})'(u_n)\|_{*}<\frac{\|u_n\|^{q}+\|u_n\|^{p}}{n (1+\|u_n\|)}\quad
\text{for all }n\in \mathbb{N}.
\end{gathered}
\end{equation}
Without loss of generality we assume that $\|u_n\|\geq 1$ for all
$n\in \mathbb{N}$. From \eqref{eq49} we have
\begin{equation} \label{eq50}
\begin{aligned}
&\Big|\langle A_{p}(u_n)+(1-t_n)A_{q}(u_n),h\rangle
 +\int_{\partial \Omega} (\beta_1(z)|u_n|^{p-2} u_n
 +(1-t_n)\beta_2(z)(u_n)^{p-2}u_n) h \,d\sigma \\
&+(1-t_n)\int_{\Omega} f(z,u_n) h\,dz
 -t_n \lambda\int_{\Omega}|u_n|^{p-2} u_n h\,dz\Big| \\
&\leq \frac{\|u_n\|^{q}+\|u_n\|^{p}}{n (1+\|u_n\|)}\\
&\leq \frac{2}{n}\|u_n\|^{p-1}
\end{aligned}
\end{equation}
for all $n\in \mathbb{N}$ (recall $\|u_n\|\geq 1$ for all
 $n\in \mathbb{N}$, $q<p$).

Let $y_n=u_n/\|u_n\|$. Then $\|y_n\|=1$ for all
$n\in \mathbb{N}$ and so we may assume that
\begin{equation}\label{eq51}
y_n\stackrel{w} \to y\text{ in }W^{1,p}(\Omega)\quad\text{and}\quad
y_n\to y\text{ in }L^{p}(\Omega)\text{ and in } L^{p}(\partial\Omega).
\end{equation}

From \eqref{eq50} we have
\begin{equation} \label{eq52}
\begin{aligned}
&\Big|\langle A_p(y_n)+\frac{1-t_n}{\|u_n\|^{p-q}} A_{q}(y_n),h\rangle  \\
&+\int_{\partial \Omega}\Big(\beta_1(z)|y_n|^{p-2} y_n
 +\frac{(1-t_n)\beta_2(z)}{\|u_n\|^{p-q}} |y_n|^{q-2} y_n\Big) h \,d\sigma \\
&-(1-t_n)\int_{\Omega}\frac{f(z,u_n)}{\|u_n\|^{p-1}} h\,dz
 -t_n\lambda\int_{\Omega}|y_n|^{p-2} y_n h\,dz\Big|\\
&\leq \frac{2}{n}\quad \text{for all }n\in \mathbb{N}.
\end{aligned}
\end{equation}
From \eqref{eq39} we have
\begin{equation}\label{eq53}
\Big\{\frac{f(\cdot,u_n(\cdot))}{\|u_n\|^{p-1}}\Big\}_{n\geq 1}
\subseteq L^{p'}(\Omega)\quad \text{is bounded}.
\end{equation}

On account of \eqref{eq53} and by passing to a subsequence if necessary and
using hypothesis (H1)(ii) we have
\begin{equation}\label{eq54}
\frac{f(\cdot,u_n(\cdot))}{\|u_n\|^{p-1}}\stackrel{w}
\to \widehat{\lambda}_m(p)|y|^{p-2}y \quad \text{in }L^{p'}(\Omega).
\end{equation}
If in \eqref{eq52} we choose $h=y_n-y\in W^{1,p}(\Omega)$, pass to the
limit as $n\to \infty$ and use \eqref{eq49}, \eqref{eq51}, \eqref{eq53}
and the fact that $q<p$, we obtain that
$\lim_{n\to \infty} \langle A_{p}(y_n),y_n-y \rangle =0$ which implies
\begin{equation}\label{eq55}
 y_n \to y\quad\text{in }W^{1,p}(\Omega),
\text{ hence $\|y\|=1$ (see Proposition \ref{prop2}).}
\end{equation}
So, if in \eqref{eq52} we pass to the limit as $n\to \infty$ and use \eqref{eq49},
\eqref{eq54}, \eqref{eq55} and the fact that $q<p$, then
\[
\langle A_p(y),h\rangle+\int_{\partial \Omega} \beta_1(z)|y|^{p-2} y h
\,d\sigma=\int_{\Omega}[(1-t)\widehat{\lambda}_m(p)+t \lambda]|y|^{p-2} y h\,dz
\]
for all $h\in W^{1,p}(\Omega)$,
which implies
\begin{equation}\label{eq56}
\begin{gathered}
 -\Delta_p y(z)=\lambda_t |y(z)|^{p-2} y(z)\quad\text{for a.a. }~z\in \Omega, \\
\frac{\partial u}{\partial n_{pq}}+\beta_1 (z)|y|^{p-2}y=0 \quad \text{on }
\partial \Omega.
\end{gathered}
\end{equation}
where $\lambda_t=(1-t)\widehat{\lambda}_m(p)+t\lambda$.

If $\lambda\notin \widehat{\sigma}(p)$, then from \eqref{eq56} it follows that
$y=0$, which contradicts \eqref{eq55}. So, suppose that
$\lambda_t\in \widehat{\sigma}(p)$. Since $y\neq 0$ (see \eqref{eq55}),
we can find $D\subseteq \Omega$ measurable with $|D|_{N}>0$ such that
\begin{equation}\label{eq57}
|u_n(z)|\to +\infty\quad\text{for all }z\in \Omega.
\end{equation}

From \eqref{eq13} we see that we can find $c_{13}>0$ and $M>0$ such that
\begin{equation}\label{eq58}
0<c_{13}\leq \frac{f_0(z,x)x-p F_0(z,x)}{|x|^{p-1}}\quad \text{for a.a. }
z\in \Omega,\text{ all }|x|\geq M.
\end{equation}
From \eqref{eq57}, \eqref{eq58} and Fatou's lemma, we have
\begin{equation}\label{eq59}
0<\liminf_{n\to \infty}\int_{\Omega}\frac{f_0(z,u_n)u_n
-pF_0(z,u_n)}{|u_n|^{p-1}}|y_n|^{p-1}\,dz.
\end{equation}
Note that on account of \eqref{eq49}, we can find $n_0\in \mathbb{N}$ such that
\begin{equation}\label{eq60}
\gamma_p(u_n)+(1-t_n)\frac{p}{q}\gamma_{q}(u_n)-(1-t_n)
\int_{\Omega} p F(z,u_n)\,dz-t_n\lambda \|u_n\|_p^p\leq 0
\end{equation}
for all $n\geq n_0$.

Also, using \eqref{eq50} with $h=u_n\in W^{1,p}(\Omega)$, we obtain
\begin{equation}\label{eq61}
-\gamma_p(u_n)-(1-t_n)\gamma_{q}(u_n)+(1-t_n)\int_{\Omega}
f(z,u_n) u_n\,dz+t_n\lambda\|u_n\|_p^p\leq \frac{2}{n}\|u_n\|^{p-1}
\end{equation}
for all$n\in \mathbb{N}$.
Adding \eqref{eq60} and \eqref{eq61} and recalling that $q<p$, we obtain
\[
(1-t_n)\int_{\Omega}[f(z,u_n) u_n-p F(z,u_n)]\,dz
\leq \frac{2}{n}\|u_n\|^{p-1}\quad\text{for all }n\geq n_0,
\]
which implies
\[
(1-t_n)\int_{\Omega}[f_0(z,u_n) u_n- pF_0(z,u_n)]\,dz
\leq \frac{2}{n}\|u_n\|^{p-1}\quad\text{for all }n\geq n_0.
\]
Evidently $t<1$, otherwise from \eqref{eq56} $y=0$ which contradicts \eqref{eq55}.
 So
\[
\int_{\Omega} [f_0(z,u_n) u_n- p F_0(z,u_n)]\,dz
\leq \frac{2 c_{14}}{n}\|u_n\|^{p-1}
\]
for all $n\geq n_0$, some $c_{14}>0$, which implies
\begin{gather*}
 \int_{\Omega}\frac{f_0(z,u_n)u_n - p F_0(z,u_n)}{|u_n|^{p-1}}|y_n|^{p-1}\,dz
\leq \frac{2 c_{14}}{n},\\
  \limsup_{n\to \infty}\int_{\Omega}
\frac{f_0(z,u_n) u_n-p F_0(z,u_n)}{|u_n|^{p-1}}|y_n|^{p-1}\,dz\leq 0,
\end{gather*}
which contradicts \eqref{eq59}. This proves  Claim 3.

On account of  Claim 3, we can use Proposition \ref{prop4} and have
\begin{equation}\label{eq62}
C_k(\varphi,\infty)=C_k(\psi,\infty)\quad\text{for all }k\in \mathbb{N}_0.
\end{equation}

Next for $r>0$ we define the following two sets
\begin{gather*}
C_r=\{u\in W^{1,p}(\Omega):\gamma_p (u)<\lambda\|u\|_p^p,\;\|u\|=r\},\\
D=\{u\in W^{1,p}(\Omega):\gamma_p (u)\geq \lambda\|u\|_p^p\}.
\end{gather*}
The set $\partial B_r=\{u\in W^{1,p}(\Omega):\|u\|=r\}$ is a $C^1$-Banach
manifold hence it is locally contractible (see Lee \cite{Lee}).
The set $C_r$ is an open subset of $\partial B_r$, hence it is locally
contractible too. Similarly the open set $W^{1,p}(\Omega)\backslash {D}$
is locally contractible too. Since
$\lambda\in (\widehat{\lambda}_m(p),\widehat{\lambda}_{m+1}(p))
\backslash {\widehat{\sigma}(p)}$,
we have
$$
\operatorname{ind} C_{r}=\operatorname{ind}(W^{1,p}(\Omega)\backslash {D})=m.
$$
So, using \cite[Theorems 3.2 and 3.6]{CingolaniDegiovanni}, we have
$C_m(\psi,0)\neq 0$.

However, since $K_{\psi}=\{0\}$, we have
\[
C_k(\psi,0)=C_k(\psi,\infty)\quad\text{for all }k\in \mathbb{N}_0
\]
(see Motreanu-Motreanu-Papageorgiou
\cite[Proposition 6.61, p.160]{MotreanuMotreanuPap},
which implies $C_{m}(\psi,\infty)\neq 0$,
$\Rightarrow$ $C_{m}(\varphi,\infty)\neq 0$ (see \eqref{eq62}).
\end{proof}

Next we compute the critical groups at infinity for the functionals
$\widehat{\varphi}_{\pm}$.

\begin{proposition}\label{prop11}
If hypotheses {\rm (H1),  (H2)} hold, then $C_k(\widehat{\varphi}_{\pm},0)=0$
for all $k\in \mathbb{N}_0$.
\end{proposition}

\begin{proof}
As before let
$\lambda\in (\widehat{\lambda}_m(p),\widehat{\lambda}_{m+1}(p))\backslash
\widehat{\sigma}(p)$ and let
$\widehat{\psi}_+:W^{1,p}(\Omega)\to \mathbb{R}$ be the $C^1$-functional defined by
$$
\widehat{\psi}_+(u)=\frac{1}{p}\gamma_p(u)+\frac{1}{p}\|u^{-}\|_p^p
+\frac{1}{p}\int_{\partial \Omega}\beta_1 (z)|u|^p \,d\sigma
-\frac{\lambda}{p}\|u^{+}\|_p^p\;\text{for all}\;u\in W^{1,p}(\Omega).
$$
We consider the homotopy $(h_+)_t(u)$ defined by
$$
(h_+)_t(u)=(1-t)\widehat{\varphi}_+(u)+t\widehat{\varphi}_+(u)\quad
\text{for all }t\in [0,1],\;u\in W^{1,p}(\Omega).
$$
We have $(h_+)_{0}=\widehat{\varphi}_+$ which satisfies the C-condition
(see Proposition \ref{prop6}). Also $(h_+)_{1}=\widehat{\psi}_+$ and since
$\lambda\notin \widehat{\sigma}(p)$, $\widehat{\psi}_+$ satisfies the C-condition.
\smallskip

\noindent\textbf{Claim 4:}
 There exist $\theta_0\in \mathbb{R}$ and $\delta_0>0$ such that
$(h_+)_{t}(u)\leq \theta_0$ which implies
\[
 (1+\|u\|)\|((h_+)_{t})'(u)\|_{*}\geq \delta_0[\|u\|^q+\|u\|^p]\quad\text{for all }
t\in [0,1].
\]
As in previous occasions, we proceed by a contradiction argument. So,
suppose that Claim 4 is not true. Since $(t,u)\to (h_+)_{t}(u)$ maps bounded
sets to bounded sets, we can find two sequences
$\{t_n\}_{n\geq 1}\subseteq [0,1]$ and
$\{u_n\}_{n\geq 1}\subseteq W^{1,p}(\Omega)$ such that
\begin{equation} \label{eq63}
\begin{gathered}
t_n\to t,\quad \|u_n\|\to \infty,\quad
(h_+)_{t_n}(u_n)\to -\infty\quad \text{and}\\
\|((h_+)_{t_n})'(u_n)\|_{*}<\frac{\|u_n\|^q+\|u_n\|^p}{n(1+\|u_n\|)}\quad
\text{for all }n\in \mathbb{N}.
\end{gathered}
\end{equation}
From  inequality \eqref{eq63} we have
\begin{equation} \label{eq64}
\begin{aligned}
&\Big|\langle A_p (u_n),h\rangle +(1-t_n)\langle A_q(u_n),h\rangle \\
&+\int_{\partial \Omega}(\beta_1(z)|u_n|^{p-2} u_n+(1-t_n)\beta_2 (z)|u_n|^{q-2} u_n) h d \sigma-\int_{\Omega}(u_n^{-})^{p-1} h\,dz \\
&-(1-t_n)\int_{\Omega} f(z,u_n^{+}) h\,dz
 -t_n\lambda \int_{\Omega}(u_n^{+})^{p-1} h\,dz\Big| \\
&\leq \frac{\varepsilon_n\|h\|}{(1+\|u_n\|}\quad
\text{for all }h\in W^{1,p}(\Omega)\text{ with }\varepsilon_n\to 0^+.
\end{aligned}
\end{equation}
Choosing  $h=-u_n^{-}\in W^{1,p}(\Omega)$ in the above inequality, we have
\begin{equation} \label{eq65}
\gamma_{p}(u_n^{-})+(1-t_n)\gamma_{q}(u_n^{-})+\|u_n^{-}\|_p^p
\leq \varepsilon_n\quad \text{for all } n\in \mathbb{N}
\end{equation}
see \eqref{eq14}, which implies
$u_n^{-}\to 0$ in $W^{1,p}(\Omega)$ (see hypothesis (H2) and recall
 $\gamma_q\geq 0$).
From \eqref{eq63} we know that $\|u_n\|\to \infty$. Hence \eqref{eq65}
implies that
\begin{equation}\label{eq66}
\|u_n^{+}\|\to +\infty\quad \text{as }n\to \infty.
\end{equation}
Let $y_n=\frac{u_n^{+}}{\|u_n^{+}\|}, \;n\in \mathbb{N}$.
Then $\|y_n\|=1$, $y_n\geq 0$ for all $n\in \mathbb{N}$. We may assume that
\begin{equation}\label{eq67}
y_n\stackrel{w} \to y\text{ in } W^{1,p}(\Omega)\quad\text{and}\quad
y_n\to y\text{ in } L^{p}(\Omega)\text{ and in }L^{p}(\partial \Omega).
\end{equation}
From \eqref{eq64} and \eqref{eq65}, we infer that
\begin{equation} \label{eq68}
\begin{aligned}
&\Big|\langle A_p (y_n),h\rangle +\frac{1-t_n}{\|u_{p}^{+}\|^{p-q}}
\langle A_q(y_n),h\rangle \\
&+\int_{\partial \Omega}(\beta_1(z)|y_n|^{p-2} y_n
 +\frac{(1-t_n)\beta_2 (z)}{\|u_n^{+}\|^{p-q}}|y_n|^{q-2} y_n) h\,dz \\
&-(1-t_n)\int_{\Omega} \frac{f(z,u_n^{+})}{\|u_n^{+}\|^{p-1}} h\,dz
-t_n\lambda \int_{\Omega}y_n^{p-1} h\,dz\Big| \\
&\leq \varepsilon'_{n}\|h\|\quad \text{for all }
h\in W^{1,p}(\Omega)\text{ with }\varepsilon'_n\to 0^+.
\end{aligned}
\end{equation}
In this inequality we choose $h=y_n-y\in W^{1,p}(\Omega)$, pass to the limit
as $n\to \infty$ and use \eqref{eq66}, \eqref{eq67} and the fact that $q<p$. Then
\[
\lim_{n\to \infty}\langle A_{p}(y_n),y_n-y\rangle =0
\]
which implies
\begin{equation}\label{eq69}
y_n\to y\quad \text{in }W^{1,p}(\Omega),
\text{ hence $\|y\|=1$, $y\geq 0$ (see Proposition \ref{prop2})}
\end{equation}

As before (see the proof of Proposition \ref{prop10}), hypothesis (H1)(ii)
 implies that
\begin{equation}\label{eq70}
\frac{f(\cdot,u_n^{+}(\cdot))}{\|u_n^{+}\|^{p-1}}\stackrel{w}
\to \widehat{\lambda}_{m}(p)y^{p-1}\quad\text{in }
L^{p'}(\Omega)\text{ as }n\to \infty.
\end{equation}
So, if in \eqref{eq68} we pass to the limit as $n\to \infty$ and use \eqref{eq66},
\eqref{eq69}, \eqref{eq70} and the fact that $q<p$, then
\[
\langle A_{p}(y),h\rangle+\int_{\partial \Omega} \beta(z) y^{p-1} h \,d\sigma
=\int_{\Omega}[(1-t)\widehat{\lambda}_{m}(p)+t \lambda]y^{p-1}h\,dz
\]
which implies
\begin{equation} \label{eq71}
\begin{gathered}
 -\Delta_p y(z)=\lambda_t y(z)^{p-1}\quad\text{for a.a. }z\in \Omega, \\
\frac{\partial y}{\partial n_{p}}+\beta(z)y^{p-1}=0 \quad \text{on }\partial \Omega.
\end{gathered}
\end{equation}
where $\lambda_{t}=(1-t)\widehat{\lambda}_{m}(p)+t\lambda$.

If $\lambda_t\notin \widehat{\sigma}(p)$, then from \eqref{eq71} we have $y=0$,
contradicting \eqref{eq69}.

If $\lambda_t\in \widehat{\sigma}(p)$, then since
$\lambda_t\geq \widehat{\lambda}_m(p)$ and $m\geq 2$, from \eqref{eq71}
 we infer that $y$ must be nodal, contradicting \eqref{eq89}.
This proves Claim 4. Using Claim 4 and Proposition \ref{prop4} and we have
\begin{equation}\label{eq72}
C_k(\widehat{\varphi}_+,\infty)
=C_k(\widehat{\psi}_+,\infty)\quad\text{for all }k\in \mathbb{N}_0.
\end{equation}

Now we introduce the  homotopy
$$
(\widehat{h}_+)_t (u)=\widehat{\psi}_+(u)-t\int_{\Omega} u\,dz\quad
\text{for all }t\in [0,1],\text{ and }u\in W^{1,p}(\Omega).
$$
\smallskip

\noindent\textbf{Claim 5:}
 $((\widehat{h}_+)_t)'(u)\neq 0$ for all $t\in [0,1]$, all
$u\in W^{1,p}(\Omega)$, $u\neq 0$.

Note that for $t\in [0,1]$ we have
\begin{equation}\label{eq73}
((\widehat{h}_+)_t)'=\widehat{\psi}'_+(u)-t\eta^*\quad \text{in }W^{1,p}(\Omega)^*,
\end{equation}
with $\eta^*\in W^{1,p}(\Omega)^*$ such that
$\langle \eta^*,v\rangle =\int_{\Omega} v\,dz$ for all $v\in W^{1,p}(\Omega)$.
Also $K_{\widehat{\psi}_+}=\{0\}$. Indeed, if $u\in K_{\widehat{\psi}_+}$, then
$\widehat{\psi}'_+(u)=0$ which implies
\[
\langle A_{p}(u),h \rangle -\int_{\Omega}(u^{-})^{p-1} h\,dz
+\int_{\partial \Omega}\beta_1(z)|u|^{p-2} u h\,dz
=\lambda\int_{\Omega} (u^{+})^{p-1} h\,dz
\]
for all $h\in W^{1,p}(\Omega)$.
Choosing $h=-u^{-}\in W^{1,p}(\Omega)$, we obtain
\[
\|Du^{-}\|_p^p+\|u^{-}\|_p^p\leq 0
\]
(see hypothesis (H2)), which implies $ u\geq 0$.
Then we have
\begin{gather*}
 -\Delta_p u(z)=\lambda u(z)^{p-1}\quad\text{for a.a. } z\in \Omega, \\
\frac{\partial u}{\partial n_{p}}+\beta (z)u^{p-1}=0 \quad \text{on }\partial \Omega.
\end{gather*}
which implies $u=0$ since $\lambda\notin\widehat{\sigma}(p)$.
Therefore $K_{\widehat{\psi}_+}=\{0\}$ and from this and \eqref{eq73},
Claim 5 follows.

The homotopy invariance property of singular homology groups implies that for
$\rho>0$ small,
\begin{equation} \label{eq74}
\begin{aligned}
&H_k((\widehat{h}_+)_0(\cdot)^0\cap B_{\rho},(\widehat{h}_+)_{0}
 (\cdot)^0\cap B_{\rho}\backslash\{0\}) \\
&= H_k((\widehat{h}_+)_1(\cdot)^0\cap B_{\rho},
(\widehat{h}_+)_{1}(\cdot)^0\cap B_{\rho}\backslash\{0\})\quad
\text{for all }k\in \mathbb{N}_0.
\end{aligned}
\end{equation}
Using Claim 5 and  Motreanu-Motreanu-Papageorgiou
\cite[Corollary 5.35, p.115 and Corollary 6.15, p.145]{MotreanuMotreanuPap},
we have
\begin{equation}\label{eq75}
H_k((\widehat{h}_+)_1(\cdot)^0\cap B_{\rho}\;,\;(\widehat{h}_+)_{1}
(\cdot)^0\cap B_{\rho}\backslash\{0\})=0\quad\text{for all }k\in \mathbb{N}_0.
\end{equation}
On the other hand from the definition of critical groups, we have
\begin{equation}\label{eq76}
H_k((\widehat{h}_+)_0(\cdot)^0\cap B_{\rho},\;(\widehat{h}_+)_{0}(\cdot)^0
\cap B_{\rho}\backslash\{0\})=C_k(\widehat{\psi}_+,0)\quad
\text{for all }k\in \mathbb{N}_0.
\end{equation}
From \eqref{eq74}, \eqref{eq75} and \eqref{eq76}, we have
\begin{equation}\label{eq77}
C_k(\widehat{\psi}_+,0)=0\quad\text{for all }k\in \mathbb{N}_0.
\end{equation}
But recall that $K_{\widehat{\psi}_+}=\{0\}$. Therefore
\[
C_k(\widehat{\psi}_+,\infty)=C_k(\widehat{\psi}_+,0)
\quad\text{for all }k\in \mathbb{N}_0
\]
which implies
$ C_n(\widehat{\psi}_+,\infty)=0$ for all
$k\in \mathbb{N}_0$ (see \eqref{eq77});
this in turn implies
$ C_k(\widehat{\varphi}_+,\infty)=0$ for all
$k\in \mathbb{N}_0$ (see \eqref{eq72}).
Similarly we show that $C_k(\widehat{\varphi}_{-},\infty)=0$ for all
$k\in \mathbb{N}_0$.
\end{proof}

From the proof of Proposition we know that the positive solution
$u_0\in D_+$ (resp. the negative solution $v_0\in -D_+$) is a critical point
of mountain pass type for the functional $\widehat{\varphi}_+$ (resp.
The functional $\widehat{\varphi}_{-}$). So, we have
\begin{equation}\label{eq78}
C_1(\widehat{\varphi}_+,u_0)\neq 0\quad\text{and}\quad
C_1(\widehat{\varphi}_{-},v_0)\neq 0,
\end{equation}
(see  Motreanu-Motreanu-Papageorgiou
\cite[Corollary 6.81, p.168]{MotreanuMotreanuPap}).
In general to describe more precisely these critical groups, we need a
Hilbert space setting and $C^2$-regularity of the functionals.
Nevertheless, here using Propositions \ref{prop10} and \ref{prop11} and
some tools from Algebraic Topology (Homological Algebra), we are able to
compute exactly the critical groups of the energy functional $\varphi$
at $u_0$ and at $v_0$. Note that since $u_0\in D_+$, $v_0\in -D_+$ and
$\varphi'|_{C_+}=\widehat{\varphi}'_+|_{C_+}$,
$\varphi'|_{-C_+}=\widehat{\varphi}'_{-}|_{-C_+}$, (see \eqref{eq14}) we
have $u_0,v_0\in K_{\varphi}$. We assume that
$K_{\widehat{\varphi}_+}=\{0,u_0\}$, $K_{\widehat{\varphi}_{-}}=\{0,v_0\}$
or otherwise we are done.

\begin{proposition}\label{prop12}
If hypotheses {\rm (H1)--(H3)} hold and $u_0\in D_+$, $v_0\in -D_+$
are the two constant sign solutions of \eqref{eq1} produced in
Propositon \ref{prop9}, then
$C_k(\varphi,u_0)=C_k(\varphi,v_0)=\delta_{k,1}\mathbb{Z}$ for all
$k\in \mathbb{N}_0$.
Here $\delta_{k,m}$ denotes the Kronecker symbol defined by
\[
\delta_{k,m}= \begin{cases}
1 & \text{if } k=m\\
0 & \text{if } k\neq m\,.
\end{cases}
\]
\end{proposition}

\begin{proof}
We will do the proof for the pair $\{\varphi,u_0\}$, the proof for the pair
$\{\varphi,v_0\}$ being similar.

Let $\eta\leq 0<\alpha<\widehat{m}^{+}_{\rho}$ (see \eqref{eq46}).
 We consider the following triple of sets
$$
\widehat{\varphi}_+^{\eta}\subseteq \widehat{\varphi}_+^{\alpha}
\subseteq W^{1,p}(\Omega).
$$
To this triple corresponds a long exact sequence of singular homology groups
(see Motreanu-Motreanu-Papageorgiou
\cite[Proposition 6.14, p.143]{MotreanuMotreanuPap}). So, we have
\begin{equation}\label{eq79}
\cdots \to H_k(W^{1,p}(\Omega),\widehat{\varphi}_+^{\eta})\stackrel{i_{*}}
\to H_k(W^{1,p}(\Omega),\widehat{\varphi}_+^{\alpha})
\stackrel{\widehat{\theta}_{*}} \to H_k(\widehat{\varphi}_+^{\alpha},
\widehat{\varphi}_+^{\eta})\to \cdots
\end{equation}
with $i_{*}$ being the homomorphism induced by the inclusion
$i:(W^{1,p}(\Omega),\widehat{\varphi}_+^{\eta})\to
(W^{1,p}(\Omega),\widehat{\varphi}_+^{\alpha})$ and $\widehat{\theta}_{*}$
is the composed boundary homomorphism. From the rank theorem we have
\begin{equation} \label{eq80}
\begin{aligned}
\operatorname{rank} H_k(W^{1,p}(\Omega),\widehat{\varphi}_+^{\alpha})
&=\operatorname{rank}\ker \widehat{\theta}_{*}
 +\operatorname{rank}\operatorname{im}\widehat{\theta}_{*} \\
&=\operatorname{rank}\operatorname{im}i_{*}
+\operatorname{rank}\operatorname{im}\widehat{\theta}_{*}
\quad \text{(since \eqref{eq79} is exact)}
\end{aligned}
\end{equation}
Since $\alpha\in (0,\widehat{m}^+_{\rho})$ and $K_{\widehat{\varphi}_+}=\{0,u_0\}$,
 we have
\begin{equation}\label{eq81}
H_k(W^{1,p}(\Omega)\widehat{\varphi}_+^{\alpha})
=C_k(\widehat{\varphi}_+,u_0)\quad\text{for all }k\in \mathbb{N}_0,
\end{equation}
(see Motreanu-Motreanu-Papageorgiou \cite[Lemma 6.55, p.157]{MotreanuMotreanuPap}).

Similarly since $\eta<0=\widehat{\varphi}_+(0)$, we have
\[
H_k(W^{1,p}(\Omega),\widehat{\varphi}_+^{\eta})
=C_k(\widehat{\varphi}_+,\infty)\quad\text{for all }k\in \mathbb{N}_0\quad
\text{(recall $K_{\widehat{\varphi}_+}=\{0,u_0\}$)}
\]
which implies
$H_k(W^{1,p}(\Omega),\widehat{\varphi}_+^{\eta})=0$ for all
$k\in \mathbb{N}_0$ (see Proposition \ref{prop11}).
This in turn implies
\begin{equation} \label{eq82}
\operatorname{im} i_{*}=\{0\}\quad \text{(see \eqref{eq79})}.
\end{equation}
In a similar fashion, we see that
$H_{k-1}(\widehat{\varphi}_+^{\alpha},\widehat{\varphi}_+^{\eta})
=C_{k-1}(\widehat{\varphi}_+,0)$ for all $k\in \mathbb{N}_0$,
which implies
\begin{equation} \label{eq83}
H_{k-1}(\widehat{\varphi}_+^{\alpha},\widehat{\varphi}_+^{\eta})
=\delta_{k-1,0}\mathbb{Z}=\delta_{k,1}\mathbb{Z}\quad\text{for all }
k\in \mathbb{N}_0\quad\text{(see Proposition \ref{prop9})}.
\end{equation}
We return to \eqref{eq80} and use \eqref{eq81}, \eqref{eq82}, \eqref{eq83}. Then
\begin{equation}\label{eq84}
\operatorname{rank} C_k(\widehat{\varphi}_+,u_0)\leq 1.
\end{equation}

Note that on account of \eqref{eq83} only the tail of \eqref{eq79} is nontrivial
 (that is, the terms for $k\geq 2$ are all zero). So, from \eqref{eq84} and
\eqref{eq78} it follows that
\begin{equation}\label{eq85}
C_k(\widehat{\varphi}_+,u_0)=\delta_{k,1}\mathbb{Z}\quad\text{for all }
k\in \mathbb{N}_0.
\end{equation}
Next consider the homotopy $\tilde{h}_t(u)$ defined by
$$
\tilde{h}_t(u)=(1-t)\varphi(u)+t\widehat{\varphi}_t(u)\quad\text{for all }
t\in [0,1],\;\text{all}\;u\in W^{1,p}(\Omega).
$$
Suppose that we could find $\{t_n\}_{n\in \mathbb{N}}\subseteq [0,1]$
and $\{u_n\}_{n\in \mathbb{N}}\subseteq W^{1,p}(\Omega)$ such that
\begin{equation}\label{eq86}
t_n\to t\text{ in }[0,1],\quad
u_n\to u_0\text{ in }W^{1,p}(\Omega)\text{ and }
(\tilde{h}_{t_n})(u_n)=0\quad\text{for all }n\in \mathbb{N}.
\end{equation}
From  equality \eqref{eq86} we have
\begin{align*}
&\langle A_p(u_n), h\rangle +\langle A_{q}(u_n), h\rangle
+\int_{\partial \Omega}(\beta_1(z)|u_n|^{p-2}u_n+\beta_2(z)|u_n|^{q-2}u_n) h \,d\sigma\\
&-t_n\int_{\Omega} (u_n^{-})^{p-1} h\,dz\\
&=(1-t_n)\int_{\Omega} f(z,u_n) h\,dz
+t_n\int_{\Omega} f(z,u_n^{+}) h\,dz\quad\text{for all }h\in W^{1,p}(\Omega),
\end{align*}
which implies
\begin{gather*}
 -\Delta_p u_n(z)-\Delta_q u_n(z)=(1-t_n)f(z,u_n(z))
+t_n[f(z,u_n^{+}(z))+u_n^{-}(z)]\\
\text{for a.a. }z\in \Omega, \\
\frac{\partial u}{\partial n_{pq}}
+\beta_1 (z)|u_n|^{p-2}u_n+\beta_2(z)|u_n|^{q-2}u_n=0 \quad \text{on }\partial \Omega.
\end{gather*}
From Papageorgiou-Radulescu \cite{PaRa5}, we know that we can find $M_4>0$ such that
$$
\|u_n\|_{\infty}\leq M_4\quad\text{for all }n\in \mathbb{N}.
$$
Then the nonlinear regularity theory of Lieberman \cite{aLieberman1991} implies
that we can find $\alpha\in (0,1)$ and $M_5>0$ such that
\begin{equation}\label{eq87}
u_n\in C^{1,\alpha}(\overline{\Omega})\quad\text{and}\quad
\|u_n\|_{C^{1,\alpha}(\overline{\Omega})}\leq M_5\quad\text{for all }n\in \mathbb{N}.
\end{equation}
The compact embedding of $C^{1,\alpha}(\overline{\Omega})$ onto
$C^1(\overline{\Omega})$ and \eqref{eq66} imply that
$u_n\to u_0$ in $C^1(\overline{\Omega})$, thus
\[
u_n\in D_+\quad\text{for all }n\geq n_0\quad \text{(recall that $u_0\in D_+$)}.
\]
But as we already pointed out
$\varphi'|_{C_+}=\widehat{\varphi}'_+|_{C_+}$ (see \eqref{eq14}) implies that
$\{u_n\}_{n\geq n_0}$ are distinct positive solutions of \eqref{eq1},
a contradiction
(recall $K_{\widehat{\varphi}_+}=\{0,u_0\}$).
So \eqref{eq86} can not occur and from the homotopy invariance of critical
groups (see for example Gasinski-Papageorgiou \cite[Theorem 5.125, p.836]{GPap3}),
we have
\begin{gather*}
C_k(\varphi,u_0)=C_k(\widehat{\varphi}_+,u_0)\quad\text{for all }k\in \mathbb{N}_0,\\
\Rightarrow  C_k(\varphi,u_0)=\delta_{k,1}\mathbb{Z}\quad\text{for all }
k\in \mathbb{N}_0\quad\text{(see \eqref{eq85})}.
\end{gather*}
Similarly, using this time the functional $\widehat{\varphi}_{-}$,
we show that
$$C_k(\varphi,v_0)=\delta_{k,1}\mathbb{Z}\quad\text{for all }k\in \mathbb{N}_0.
$$
\end{proof}

Now we can produce a third nontrivial smooth solution for problem \eqref{eq1}
and formulate our ``three solutions theorem''.

\begin{theorem}\label{thm13}
If hypotheses {\rm (H1)--(H3)} hold, then problem \eqref{eq1} admits at least
three nontrivial smooth solutions
$$
u_0\in D_+,\quad v_0\in -D_+, \quad y_0\in C^1(\overline{\Omega}).
$$
\end{theorem}

\begin{proof}
From Proposition \ref{prop9}, we already have two constant sign solutions
$$
u_0\in D_+,\quad v_0\in -D_+.
$$
From Proposition \ref{prop12} we know that
\begin{equation}\label{eq88}
C_k(\varphi,u_0)=C_k(\varphi,v_0)=\delta_{k,1}\mathbb{Z}\quad
\text{for all }k\in \mathbb{N}_0.
\end{equation}
Also, from Proposition \ref{prop8}, we know that $u=0$ is a local minimizer
of $\varphi$. Hence
\begin{equation}\label{eq89}
C_k(\varphi,0)=\delta_{k,0}\mathbb{Z}\quad\text{for all }k\in \mathbb{N}_0.
\end{equation}
From Proposition \ref{prop10}, we have that $C_m(\varphi,\infty)\neq 0$
($m\geq 2$). So, we can find $y_0\in K_{\varphi}$ such that
$C_m(\varphi,y_0)\neq 0$ and $y_0\notin \{0,u_0,v_0\}$
(see Motreanu-Motreanu-Papageorgiou
\cite[Proposition 6.89, p.172]{MotreanuMotreanuPap}).
Then $y_0$ is a third nontrivial solution of \eqref{eq1} and the nonlinear
regularity theory of Lieberman \cite{aLieberman1991} implies that
$y_0\in C^1(\overline{\Omega})$.
\end{proof}

\subsection*{Acknowledgements}
This research was partly supported by the University of Piraeus
Research Center.

The authors wish to thank the anonymous referee for reading the paper
carefully and giving us corrections and remarks.
Also they thank Professor Otani for remarks on the revised version.


\begin{thebibliography}{99}

\bibitem{AizicPapStaicu1} S. Aizicovici, N. S. Papageorgiou, V. Staicu;
\emph{Degree Theory for Operators of Monotone Type and Nonlinear
Elliptic Equations with Inequality Constraints}, Memoirs of the
AMS, Vol. 196, No \textbf{915} (2008), pp. 70.

\bibitem{AizicPapStaicu2} S. Aizicovici, N. S. Papageorgiou, V.Staicu;
\emph{Nodal solutions for $(p,2)$-equations},
Trans AMS, \textbf{367} (2015), 7343-7372.

\bibitem{AmbRab}  A. Ambrosetti, P. Rabinowitz;
 \emph{Dual variational methods in critical point theory and applications},
J. Functional Anal., \textbf{14} (1973), 349-381.

\bibitem{BenAvenFortPis}  V. Benci, P. D'Avenia, D. Fortunato, L. Pisani;
 \emph{Solitons in several space dimensions: Derrick'sproblem and infinitely
many solutions}, Arch. Rat. Mech. Anal. \textbf{154} (2000), 297-324.

\bibitem{BenBel}  N. Benouhiba, Z. Belyacine;
\emph{On the solutions of the $(p,q)$-Laplacian problem at resonance},
Nonlin. Anal., \textbf{77} (2013), 74-81.

\bibitem{BobkovTanaka}  V. Bobkov, M. Tanaka;
 \emph{On positive soluitons for $(p,q)$-Laplace equations with two parameters},
Calc. Var., \textbf{54} (2015), 1959-1972.

\bibitem{CherfilsILyason}  L. Cherfils, Y. Ilyason;
 \emph{On the stationary solutions of generalized reaction-diffusion equations
with $(p,q)$-Laplacian},
Comm. Pure Appl. Anal., \textbf{4} (2005), 9-22.

\bibitem{CingolaniDegiovanni}  S. Cingolani, M. Degiovanni;
 \emph{Nontrivial solutions for the $p$-Laplace equations with right hand
side having $p$-linear growth at infinity},
Comm. Partial Diff. Equations,  \textbf{30} (2005), 1191-1203.

\bibitem{GPap1}  L. Gasinski, N. S. Papageorgiou;
\emph{Nonlinear Analysis},
 Chapman and Hall/CRC Press, Boca Raton, (2006).

\bibitem{GPap2} L. Gasinski, N. S. Papageorgiou;
 \emph{Dirichlet $(p,q)$-equations at resonance},
Discr. Cont. Dynam. Systems A, \textbf{34} (2014), 2037-2060.

\bibitem{GPap3}  L. Gasinski, N. S. Papageorgiou;
 \emph{Exercises in Analysis Part 2:Nonlinear Analysis},
 Springer Cham, Heidelberg, (2016)

\bibitem{Lee}  J. M. Lee;
 \emph{Introduction to Topological Manifolds},
 Springer-Verlag, New York (2000).

\bibitem{LiangSu} Z. Liang-J. Su;
 \emph{Multiple solutions for semilinear elliptic boundary value problems
 with double resonance}, J. Math. Anal. Appl. \textbf{354} (2009), 147-158.

\bibitem{aLieberman1991}  G. M. Lieberman;
 \emph{The natural generalization of the natural conditions of Ladyzhenskaya
and Uraltseva for elliptic equations},  Comm. Partial Diff. Equations
 16 (1991), 311-361.

\bibitem{MaranoMosconiPap}  S. Marano-S. Mosconi, N. S. Papageorgiou,
 \emph{Multiple solutions to $(p,q)$-Laplacian problem with resonant concave
nonlinearity}, Adv. Nonlin. Studies \textbf{16} (2016), 51-65.

\bibitem{MaranoPap}  S. Marano-N. S. Papageorgiou;
 \emph{Constant sign and nodal solutions of concave $(p,q)$-Laplacian problems},
Nonlin. Anal., \textbf{77} (2013), 118-129.

\bibitem{MotreanuMotreanuPap}  D. Motreanu, V. Motreanu-N. S. Papageorgiou;
 \emph{Topological and Variationsl Methods with Applications to Nonlinear
Boundary Value Problems}, Springer, New York (2014).

 \bibitem{MugnaiPap1}  D. Mugnai, N. S. Papageorgiou;
 \emph{Resonant nonlinear Neumann problems with indefinite weight},
Annali Scuola Normale Super. Pisa CI. Sci II, \textbf{5} (2012), 729-788.

 \bibitem{MugnaiPap2}
 D. Mugnai, N. S. Papageorgiou;
 \emph{Wang's multiplicity result for superlinear $(p,q)$-equations
without the Ambrosetti-Rabinowitz condition},
Trans. AMS,  \textbf{366} (2014), 4919-4937.

 \bibitem{PaRa1}  N. S. Papageorgiou, V. D. Radulescu;
 \emph{Quatitative phenomena for some clasees of quasilinear elliptic equations
with multiple resonance},
 Appl. Math. Optim., \textbf{69} (2014), 393-430.

 \bibitem{PaRa2}  N. S. Papageorgiou, V. D. Radulescu;
 \emph{Solutions with sign information for nonlinear nonhomogeneous elliptic
equations},  Topol, Metho. Nonlin. Anal., \textbf{45} (2015), 575-600.

 \bibitem{PaRa3}  N. S. Papageorgiou, V. D. Radulescu;
 \emph{Multiplicity theorem for nonlinear nonhomogeneous Robin problem},
Revista Mat. Iberoamericana, \textbf{33} (2017), 251-289.

\bibitem{PaRa4}  N. S. Papageorgiou, V. D. Radulescu;
 \emph{Multiple solutions with precise sign for parametric Robin problems},
 J. Differential Equations, \textbf{256} (2014), 2449--2479.

\bibitem{PaRa5}
 N. S. Papageorgiou, V. D. Radulescu;
 \emph{Nonlinear nonhomogeneous Robin problems with superlinear reaction term},
 Adv. Nonlin. Studies, \textbf{16} (2016), 737-764.

\bibitem{PaWinkert}  N. S. Papageorgiou, P. Winkert,
 \emph{Resonant $(p,2)$-equations with concave terms},
 Appl. Anal. $\textbf{94}$ (2015), 341-359.

 \bibitem{PuSe} P. Pucci, J. Serrin;
\emph{The maximum Principle}, Birkh\"{a}user, Basel, 2007.

\bibitem{Sun}  M. Sun;
 \emph{Multiplicity of solutions for a class of quasilinear elliptic equations
 with resonance}, J. Math. Anal. Appl., \textbf{386} (2012), 661-668.

\bibitem{SunZhangSu}  M. Sun, M. Zhang, J. Su;
 \emph{Critical groups of zero and multiple solutions for a quasilinear elliptic
 equation}, J. Math. Anal. Appl., \textbf{428} (2015), 712.

\bibitem{Tanaka}  M. Tanaka;
 \emph{Uniqueness of a positive solution and existence of a sign-changing
solution for $(p,q)$-Laplacian equation},
J. Nonlin. Funct. Anal., No. \textbf{14} (2014), pp.15.

\bibitem{YangYin}  Z. Yang,-H. Yin;
 \emph{A class of $(p,q)$-Laplacian type equation with concave-convex
nonlinearities in bounded domain},
J. Math. Anal. Appl., \textbf{382} (2011), 843-855.

\end{thebibliography}

\end{document}