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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 182, pp. 1--14.\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-2018/182\hfil Nonlinear Robin problems]
{Nonlinear Robin problems with unilateral constraints and dependence\\
 on the gradient}

\author[N. S. Papageorgiou, C. Vetro, F. Vetro \hfil EJDE-2018/182\hfilneg]
{Nikolaos S. Papageorgiou, Calogero Vetro, Francesca Vetro}

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

\address{Calogero Vetro (corresponding author) \newline
University of Palermo,
Department of Mathematics and Computer Science,
Via Archirafi 34, 90123,
Palermo, Italy}
\email{calogero.vetro@unipa.it}

\address{Francesca Vetro \newline
Nonlinear Analysis Research Group,
Ton Duc Thang University,
Ho Chi Minh City, Vietnam. \newline
Faculty of Mathematics and Statistics,
Ton Duc Thang University, Ho Chi Minh City, Vietnam}
\email{francescavetro@tdtu.edu.vn}

\dedicatory{Communicated by Marco Squassina}

\thanks{Submitted December 17, 2017. Published November 13, 2018.}
\subjclass[2010]{35J20, 35J60, 35J92}
\keywords{$p$-Laplacian; Robin boundary condition; subdifferential term;
\hfill\break\indent convection term; nonlinear regularity;
 maximal monotone map; fixed point}

\begin{abstract}
 We consider a nonlinear Robin problem driven by the $p$-Laplacian,
 with unilateral constraints and a reaction term depending also on
 the gradient (convection term). Using a topological approach based on
 fixed point theory (the Leray-Schauder alternative principle) and
 approximating the original problem using the Moreau-Yosida approximations
 of the subdifferential term, we prove the existence of a smooth solution.
\end{abstract}

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


\section{Introduction}

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 elliptic
differential inclusion with Robin boundary condition
\begin{equation}\label{eq1}
\begin{gathered}
- \Delta_p u(z) + \partial \varphi(u(z)) \ni  f(z,u(z), \nabla u(z))\quad
  \text{in }\Omega, \\
\quad \frac{\partial u}{\partial n_p} + \beta (z)|u|^{p-2}u=0 \quad
  \text{on }\partial \Omega.
\end{gathered}
\end{equation}
In this problem $\Delta_p$ denotes the $p$-Laplace differential operator defined by
$$
\Delta_p u = \operatorname{div}(|\nabla u|^{p-2} \nabla u) \quad \text{for all } 
u \in W^{1,p}(\Omega), \quad 2 \leq p < +\infty.
$$

Also $\varphi \in \Gamma_0(\mathbb{R})$, the cone of proper, convex and lower 
semicontinuous functions (see Section \ref{Sec2}),
and $\partial \varphi(\cdot)$ denotes the subdifferential in the sense of 
convex analysis. The presence of the subdifferential term, incorporates 
in our framework problems with unilateral constraints (differential variational 
inequalities). The forcing term $f(z,x,y)$ is a measurable function which is 
locally H\"{o}lder in the $(x,y) \in \mathbb{R} \times \mathbb{R}^N$ variables. 
The dependence on the gradient implies that we can not use directly on 
\eqref{eq1} variational methods. For this reason our approach is topological 
based on the fixed point theory (the Leray-Schauder alternative principle). 
In the boundary condition, $\frac{\partial u}{\partial n_p}$ is the conormal 
derivative of $u$ defined by extension of the map
$$
C^1(\overline{\Omega}) \ni u \to |\nabla u|^{p-2} \frac{\partial u}{\partial n},
$$
with $n(\cdot)$ being the outward unit normal on $\partial \Omega$.

By a solution of \eqref{eq1} we understand a function $u \in W^{1,p}(\Omega)$
such that there exists $g \in L^2(\Omega)$ satisfying 
$g(z) \in \partial \varphi (u(z))$ for a.a.\ $z \in \Omega$ and
$$
\int_\Omega |\nabla u |^{p-2}(\nabla u ,\nabla h)_{\mathbb{R}^N}\,dz 
+ \int_{\partial \Omega} \beta(z)|u|^{p-2} u h d\sigma +\int_\Omega gh\,dz
=\int_\Omega f(z,u,\nabla u)h \,dz 
$$
for all $h \in W^{1,p}(\Omega)$.

Existence theorems for nonlinear elliptic equations with convection were 
proved by de Figueiredo-Girardi-Matzeu \cite{Ref2}, Girardi-Matzeu \cite{Ref5} 
(semilinear Dirichlet problems driven by the Laplacian), Ruiz \cite{Ref16}, 
Faraci-Motreanu-Puglisi \cite{Ref1}, Huy-Quan-Khanh \cite{Ref7}, 
Iturriaga-Lorca-S\'anchez \cite{Ref8} (nonlinear Dirichlet problems driven 
by the $p$-Laplacian) and Gasi\'nski-Papageorgiou \cite{Ref4}, 
Papageorgiou-R\v{a}dulescu-Repov\v{s} \cite{Ref15} (Neumann problems driven by 
a differential operator of the form $\operatorname{div} (a(u) \nabla u)$ 
with $a(\cdot)$ continuous, bounded and strictly positive). 
Of the aforementioned works, only Papageorgiou-R\v{a}dulescu-Repov\v{s} \cite{Ref15}
 consider problems with unilateral constraint (that is, with a subdifferential 
term $\partial \varphi(u)$). Their conditions on the convection term are 
different and they employ a suitable variant of the classical Nagumo-Hartman 
condition. Their differential operator is of the form 
$\operatorname{div} (a(u) \nabla u)$ with $a:\mathbb{R} \to \mathbb{R}$ 
Lipschitz continuous and $0<c_1 \leq a(x) \leq c_2$ for all $x \in \mathbb{R}$. 
This particular form of the differential operator is essential for their proofs 
to work and their method can not accommodate a nonlinear differential operator 
like the $p$-Laplacian.

The presence of the subdifferential term $\partial \varphi(u)$ introduces 
a multivalued unilateral constraint in the problem which complicates the study 
of \eqref{eq1}. Our aim is to prove an existence theorem for problem \eqref{eq1}. 
In fact we show the existence of a smooth solution for problem. 
The method of proof passes through a regularization of the subdifferential map. 
The regularization is based on the Moreau-Yosida approximation of $\varphi$. 
Exploiting the properties of the Moreau-Yosida approximation we solve the 
regularized problem, using topological tools based on the fixed point theory. 
Using the nonlinear regularity theory, we derive uniform a priori bounds for
 the solutions of the approximate problems and then we pass to the limit to 
obtain the desired solution of \eqref{eq1}.


 \section{Mathematical background and hypotheses}\label{Sec2}

 Let $X$ be a reflexive Banach space and $X^\ast$ its topological dual. 
By $\langle\cdot,\cdot \rangle$ we denote the duality brackets for the dual 
pair $(X^\ast,X)$. A map $A: D(A) \subseteq X \to 2^{X^*}$ is said to 
be ``monotone'', if
 $$
\langle x^*-u^*,x-u \rangle \geq 0 \quad \text{for all } (x,x^*), \;
 (u,u^*) \in  \operatorname{Gr}A.
$$
Here $\operatorname{Gr}A$ is the ``graph of $A(\cdot)$'' defined by
$$
\operatorname{Gr}A= \{(x,x^*) \in X \times X^* : x^* \in A(x)\}
$$ 
and $D(A)$ is the ``domain of $A(\cdot)$'' defined by
$$
D(A)=\{x \in X : A(x) \neq \emptyset\}.
$$

We say that $A(\cdot)$ is ``strictly monotone'', if it is monotone and
$$
 \langle x^*-u^*,x-u \rangle = 0 \;\Rightarrow \; x=u.
$$
A monotone map is ``maximal monotone'', if its graph is maximal among the 
graphs of monotone maps. This means that
$$
\langle u^*-x^*,u-x \rangle \geq 0  \quad \text{for all } (x,x^*)  \in  
\operatorname{Gr}A \;\Rightarrow \;  (u,u^*)  \in
 \operatorname{Gr}A.
$$

The importance of maximal monotone maps, comes from their remarkable 
surjectivity properties. More precisely we have (see, for example, 
Gasi\'nski-Papageorgiou \cite[p. 319]{Ref3}).

\begin{proposition}\label{P1}
If $A{:} X \to 2^{X^*}$ is maximal monotone and coercive (that is, 
$\|A(x)\|_* \to +\infty$ as $\|x\| \to +\infty$), then $A$ is surjective.
\end{proposition}

By $\Gamma_0(X)$ we denote the cone of all functions 
$\varphi{:} X \to \overline{\mathbb{R}}= \mathbb{R} \cup \{+\infty\}$ 
which are convex, lower semicontinuous and proper (that is, the effective domain 
of $\varphi$, ${\rm dom } \varphi=\{x \in X : \varphi(x) < +\infty\}$ is nonempty).

For $\varphi \in \Gamma_0(X)$, the subdifferential of $\varphi$ at $x \in X$, 
is defined by
$$
\partial \varphi(x)=\{x^* \in X^* : \langle x^*,h \rangle \leq \varphi(x+h)
-\varphi(x) \text{ for all } h \in X\}.
$$

Note that $\partial \varphi(x) \subseteq X^*$ is w-closed, convex and possibly empty. 
If $\varphi(\cdot)$ is continuous at $x \in X$, then $\partial \varphi(x)$ is 
nonempty, w-compact and convex. If $\varphi(\cdot)$ is G\^{a}teaux differentiable 
at $x \in X$, then 
$$
\partial \varphi(x) =\{\varphi'_G(x)\}
$$
with $\varphi'_G(x)$ being the G\^{a}teaux derivative of $\varphi$ at $x$. 
The map $\partial \varphi {:} X \to 2^{X^*}$ is maximal monotone.

If $X=H=$ a Hilbert space and $\varphi \in \Gamma_0(H)$, then for every 
$\lambda >0$, the ``Moreau-Yosida approximation'' $\varphi_\lambda$ of $\varphi$, 
is defined by
$$
\varphi_\lambda(x)=\inf [\varphi(u) + \frac{1}{2 \lambda}\|x-u\|^2 
: u \in H] \quad \text{for all } x \in H.
$$

This functional has the following properties:
\begin{itemize}
\item  $\varphi_\lambda$ is convex and ${\rm dom\, } \varphi_\lambda=H$.
\item  $\varphi_\lambda$ is Fr\'echet differentiable and the Fr\'echet derivative  
$\varphi'_\lambda$ is Lipschitz continuous with Lipschitz constant 
$\frac{1}{\lambda}$.
\item  If $\lambda_n \to 0^+$, $x_n \to x$ in $H$,
 $\varphi'_{\lambda_n}(u_n) \xrightarrow{w} x^*$ in $H^*$, then 
$x^* \in \partial \varphi(x)$.
\end{itemize}

Let $V$ and $Z$ be Banach spaces. We say that $g: V \to Z$ is ``compact'' 
if it is continuous and maps bounded sets in $V$ to relatively compact sets in $Z$.

As we already mentioned, our approach is topological and will make use of the 
``Leray-Schauder alternative principle'' 
(see Gasi\'nski-Papageorgiou \cite[p. 827]{Ref3}).

\begin{theorem}\label{T2}
If $V$ is a Banach space, $\varphi : V \to V$ is a compact map and 
$$
T(\varphi)=\{x \in V : \text{ there exists $t \in (0,1)$ such that 
$x =t \varphi(x)$}\},
$$
then either $T(\varphi)$ is unbounded or $\varphi$ has a fixed point.
\end{theorem}

The main space of our analysis is the Sobolev space $W^{1,p}(\Omega)$. 
Endowed with the norm
$$
\|u\| =[\|u\|_p^p+ \|\nabla u \|_p^p]^{\frac{1}{p}} \quad \text{for all } 
u \in W^{1,p}(\Omega),
$$
this Sobolev space becomes a separable reflexive Banach space. 
The nonlinear regularity theory will bring into play the Banach space 
$C^1(\overline{\Omega})$ and the Robin boundary condition the ``boundary'' 
Lebesgue spaces $L^r(\partial \Omega)$, $1 \leq r \leq +\infty$. 
To define the latter, on $\partial \Omega$, we consider the 
$(N-1)$-dimensional Hausdorff (surface) measure denoted by $\sigma(\cdot)$. 
Using this measure on $\partial \Omega$, we can define in the usual way 
the Lebesgue spaces $L^r(\partial \Omega)$. From the theory of Sobolev spaces,
we know that there exists a unique linear continuous map 
$\gamma_0 {:} W^{1,p}(\Omega) \to L^p(\partial \Omega)$, known as 
the ``trace map'', such that $\gamma_0(u)=u  |_{\partial \Omega}$ for all 
$u \in W^{1,p}(\Omega) \cap C(\overline{\Omega})$. Hence, we understand 
the trace map as representing the boundary values of an arbitrary Sobolev 
function $u \in W^{1,p}(\Omega)$. We know that $\gamma_0$ is a compact operator 
into $L^r(\partial \Omega)$ for $1 \leq r < \frac{p(N-1)}{N-p}$ when $N>p$ 
and into $L^r(\partial \Omega)$ for $1 \leq r < +\infty$, when $N \leq p$. Moreover,
$$
\operatorname{im} \gamma_0 = W^{\frac{1}{p'},p}(\partial \Omega) \quad 
\big(\frac{1}{p}+\frac{1}{p'}=1\big) \quad \text{and} \quad \ker 
 \gamma_0 = W_0^{1,p}(\Omega).
$$

In what follows, for 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.
We introduce the following condition on the boundary coefficient $\beta(\cdot)$:
\begin{itemize}
\item[(H1)] $\beta \in C^{0,\alpha}(\partial \Omega )$ with $\alpha \in (0,1)$ 
and $\beta(z) \geq 0$ for all $z \in \partial \Omega$.
\end{itemize}

Note that if $\beta \equiv 0$, then we recover the Neumann problem.
We consider the  nonlinear eigenvalue problem
\begin{equation} \label{eq2} 
\begin{gathered}
- \Delta_p u(z) = \widehat{\lambda} |u(z)|^{p-2}u(z)\quad  \text{in }\Omega, \\
\quad \frac{\partial u}{\partial n_p} + \beta (z)|u|^{p-2}u=0 \quad
 \text{on }\partial \Omega.
\end{gathered}
\end{equation}
We say that $\widehat{\lambda} \in \mathbb{R}$ is an eigenvalue of the Robin 
$p$-Laplacian, if problem \eqref{eq2} admits a nontrivial solution 
$\widehat{u} \in W^{1,p}(\Omega)$, known as an ``eigenfunction'' 
corresponding to the eigenvalue $\widehat{\lambda}$. 
We know that \eqref{eq2} admits a smallest eigenvalue $\widehat{\lambda}_1$ 
which has the following properties (see Papageorgiou-R\v{a}dulescu \cite{Ref13}):
\begin{itemize}
\item $\widehat{\lambda}_1 \geq 0$ (in fact, if $\beta \equiv 0$ (Neumann problem), 
 then $\widehat{\lambda}_1 =0$, while if $\beta \not \equiv 0$ then 
 $\widehat{\lambda}_1 >0$).
\item $\widehat{\lambda}_1$ is isolated in the spectrum $\widehat{\sigma}(p)$ 
 of the Robin $p$-Laplacian (that is, there exists $\varepsilon >0$ such that 
 $(\widehat{\lambda}_1, \widehat{\lambda}_1 + \varepsilon) \cap \widehat{\sigma}(p) 
 = \emptyset$).
\item $\widehat{\lambda}_1$ is simple (that is, if $\widehat{u}$, $\widehat{v}$ 
 are eigenfunctions corresponding to $\widehat{\lambda}_1$, then 
 $\widehat{u}=\xi \widehat{v}$ for some $\xi \in \mathbb{R} \setminus \{0\}$).
\item If $\gamma(u)=\|\nabla u\|_p^p + \int_{\partial \Omega}\beta(z)|u|^p d \sigma$, 
 then
\begin{equation}\label{eq3}
\widehat{\lambda}_1= \inf \big[\frac{\gamma(u)}{\|u\|_p^p} :
 u \in W^{1,p}(\Omega), u \neq 0 \big].
\end{equation}
\end{itemize}

The infimum in \eqref{eq3} is realized on the corresponding one dimensional
 eigenspace. It is clear from the above properties that the elements 
of this eigenspace do not change sign. By $\widehat{u}_1$ we denote the positive,
 $L^p$-normalized (that is, $\|\widehat{u}_1\|_p=1$), eigenfunction corresponding 
to $\widehat{\lambda}_1$. The nonlinear regularity theory of Lieberman \cite{Ref9} 
implies that $\widehat{u}_1 \in C^1(\overline{\Omega})$. Moreover, the nonlinear 
strong maximum principle (see Gasi\'nski-Papageorgiou  \cite[p. 738]{Ref3}), 
implies that $\widehat{u}_1(z)>0$ for all $z \in \overline{\Omega}$. 
Using these properties one can prove easily the following lemma 
(see Mugnai-Papageorgiou \cite[Lemma 4.11]{Ref12}).

\begin{lemma}\label{L3}
If $\vartheta \in L^\infty(\Omega)$, $\vartheta(z) \leq \widehat{\lambda}_1$ for a.a.\
 $z \in \Omega$ and this inequality is strict on a set of positive measure, 
then there exists $\widehat{c}>0$ such that
$$
\widehat{c}\|u\|^p \leq \gamma(u) - \int_\Omega \vartheta(z)|u|^p\,dz \quad 
\text{for all } u \in W^{1,p}(\Omega).
$$
\end{lemma}

We say that a function $f{:}\Omega \times \mathbb{R} \times \mathbb{R}^N
 \to \mathbb{R}$ is an $L^\infty$-locally H\"{o}lder function, if for all $\rho>0$, 
there exists $\eta_\rho \in L^\infty(\Omega)$ such that
$$
|f(z,x,y)-f(z,x',y')| \leq \eta_\rho(z) \left[|x-x'|^\mu + |y-y'|^\mu\right]
$$ 
for a.a.\ $z \in \Omega$, all $|x|$, $|x'|$, $|y|$, $|y'|\leq \rho$, with 
$0<\mu \leq 1$.

Our hypotheses on the reaction term $f(z,x,y)$ are:
\begin{itemize}
\item[(H2)] $f:\Omega \times \mathbb{R} \times \mathbb{R}^N \to \mathbb{R}$ 
is an $L^\infty$-locally H\"{o}lder function such that
\begin{itemize}
\item[(i)] $|f(z,x,y)| \leq   a(z) [1+|x|^{p-1}+|y|^{q}]$ for a.a.\
 $z \in \Omega$, all $x \in \mathbb{R}$, $y \in \mathbb{R}^N$, with
$2 \leq p$, 
\[
 2(p-1) \leq p^*=\begin{cases}\frac{Np}{N-p} & \text{if } p<N,\\
 +\infty & \text{if }N \leq p
\end{cases}
\]
 and $q=\max\{\frac{p-1}{2},1\}$;

\item[(ii)] there exists a function $\vartheta \in L^\infty(\Omega)$ such that 
$\vartheta(z) \leq \widehat{\lambda}_1$ for a.a.\
 $z \in \Omega$, the above inequality is strict on a set of positive measure, 
$\limsup_{x \to \pm \infty}\frac{f(z,x,y)}{|x|^{p-2}x}\leq \vartheta(z)$ 
uniformly for a.a.\ $z \in \Omega$, all $y \in \mathbb{R}^N$ on a bounded set.
\end{itemize}
\end{itemize}

Note that if $p \geq \frac{N}{2}$, then $2(p-1) \leq p^*$ (see hypothesis (H2)(i)).
The following function satisfies hypotheses (H2) 
(for the sake of simplicity, we drop the $z$-dependence),
$$
f(x,y)=\vartheta |x|^{p-2} x + g(x) |y|^q \quad \text{for all } x \in \mathbb{R}, 
\, y \in \mathbb{R}^N,
$$ 
with $\vartheta < \widehat{\lambda}_1$, $2 \leq p < +\infty$ with 
$2(p-1) \leq p^*$, $q =\max\{\frac{p-1}{2},1\}$ and $g:\mathbb{R} \to \mathbb{R}$  
is locally H\"{o}lder continuous and 
$\lim_{x \to \pm \infty} \frac{g(x)}{|x|^{p-2}x}=0$.

The hypotheses on the function $\varphi$ are:
\begin{itemize}
\item[(H3)]  $\varphi \in \Gamma_0(\mathbb{R})$ with $0 \in \partial \varphi(0)$.
\end{itemize}
These conditions mean that $\varphi \geq 0$ and $\varphi(0)= \inf \varphi$.


\section{Existence of solutions}\label{Sec3}

Recall that from hypothesis (H2)(ii), $\vartheta \in L^\infty(\Omega)$ and 
$\vartheta(z) \leq \widehat{\lambda}_1$ for a.a.\ $z \in \Omega$, 
$\vartheta \not \equiv \widehat{\lambda}_1$.  
Let $\lambda>0$, $g \in L^\infty(\Omega)$ and $\widehat{\xi} >\|\vartheta\|_\infty$. 
We  consider the  auxiliary nonlinear Robin problem
\begin{equation}\label{eq4}
\begin{gathered}
- \Delta_p u(z) +\widehat{\xi}|u(z)|^{p-2}u(z)+ \partial \varphi_\lambda(u(z)) =g(z)
\quad  \text{in }\Omega, \\
 \frac{\partial u}{\partial n_p} + \beta (z)|u|^{p-2}u=0 \quad 
 \text{on }\partial \Omega, \; \lambda>0.
\end{gathered}
\end{equation}

\begin{proposition}\label{P2}
If hypotheses {\rm (H1)} and {\rm (H3)} hold, then problem \eqref{eq4} has a unique 
solution $S_\lambda(g) \in C^1(\overline{\Omega})$ and the solution map 
$S_\lambda : L^\infty(\Omega) \to C^1(\overline{\Omega})$ is compact.
\end{proposition}

\begin{proof}
First we show the existence of a solution for problem \eqref{eq4}. 
So, we consider the operator $E: W^{1,p} (\Omega) \to W^{1,p} (\Omega)^*$ 
defined by
\begin{align*}
\langle E(u),h\rangle 
&=\int_\Omega |\nabla u |^{p-2}(\nabla u ,\nabla h)_{\mathbb{R}^N}\,dz 
+\widehat{\xi}\int_{\Omega} |u|^{p-2} u h \,dz\\ 
& \quad + \int_{\partial \Omega} \beta(z)|u|^{p-2} u h d\sigma 
+\int_\Omega \partial \varphi_\lambda(u)h\,dz  \quad 
\text{for all $u,h \in W^{1,p}(\Omega)$.}
\end{align*}
Clearly $E(\cdot)$ is continuous, monotone, thus maximal monotone 
(see \cite[p. 310]{Ref3}). Also, for all 
$u \in W^{1,p}(\Omega)$ we have
\begin{gather*}
\langle E(u),u \rangle 
\geq \|\nabla u \|_p^p + \widehat{\xi}\|u \|_p^p \geq c_0 \| u \|^p   
 \text{ with } c_0=\min\{\widehat{\xi},1\}\\
\Rightarrow \;   E(\cdot) \text{ is coercive};
\end{gather*}
see hypothesis (H1) and recall that $\partial \varphi_\lambda$ 
is monotone, $\partial\varphi_\lambda(0)=0)$.

Invoking Proposition \ref{P1}, we can find $\widehat{u}_\lambda \in W^{1,p}(\Omega)$ 
such that
\begin{align}
 & E(\widehat{u}_\lambda)=g  \nonumber \\
&\Rightarrow \;  \int_\Omega |\nabla \widehat{u}_\lambda |^{p-2}
 (\nabla \widehat{u}_\lambda ,\nabla h)_{\mathbb{R}^N}\,dz 
 +\widehat{\xi}\int_{\Omega} |\widehat{u}_\lambda|^{p-2} \widehat{u}_\lambda h \,dz
 + \int_{\partial \Omega} \beta(z)|\widehat{u}_\lambda|^{p-2} 
 \widehat{u}_\lambda h d\sigma \nonumber \\ 
& \quad  +\int_\Omega \partial \varphi_\lambda(\widehat{u}_\lambda)h\,dz 
 = \int_\Omega g h \,dz  \quad \text{for all $h \in W^{1,p}(\Omega)$,} \label{eq5}
\\ 
&\Rightarrow  \; \begin{cases}
- \Delta_p \widehat{u}_\lambda(z) +\widehat{\xi}|\widehat{u}_\lambda(z)|^{p-2}
\widehat{u}_\lambda(z)+ \partial \varphi_\lambda(\widehat{u}_\lambda(z)) =g(z)
 &  \text{for a.a. } z \in \Omega, \\
\frac{\partial \widehat{u}_\lambda}{\partial n_p} 
+ \beta (z)|\widehat{u}_\lambda|^{p-2}\widehat{u}_\lambda=0 
& \text{on }\partial \Omega;
\end{cases}\label{eq6}
\end{align}
see Papageorgiou-R\v{a}dulescu \cite{Ref13}.

We know that $|\partial \varphi_\lambda(x)| \leq \frac{1}{\lambda} |x|$ for all
 $x \in \mathbb{R}$ (see Hu-Papageorgiou \cite[p. 350]{Ref6}).
 Hence from \eqref{eq6} and Papageorgiou-R\v{a}dulescu \cite{Ref14}, 
we have $u_\lambda \in L^\infty(\Omega)$.
Invoking  Lieberman \cite[Theorem 2]{Ref9}, we infer that 
$$
u_\lambda \in C^1(\overline{\Omega}).
$$
So, we have proved the existence of a smooth solution for problem \eqref{eq4}. 
Next we show the uniqueness of this solution. 
So, suppose that $\widehat{v}_\lambda \in W^{1,p}(\Omega)$ is another solution. 
Again we have $\widehat{v}_\lambda \in C^1(\overline{\Omega})$ and
\begin{equation}
\begin{aligned}
&\int_\Omega |\nabla \widehat{v}_\lambda |^{p-2}(\nabla \widehat{v}_\lambda ,
 \nabla h)_{\mathbb{R}^N}\,dz
 +\widehat{\xi}\int_{\Omega} |\widehat{v}_\lambda|^{p-2} \widehat{v}_\lambda h \,dz \\
&+ \int_{\partial \Omega} \beta(z)|\widehat{v}_\lambda|^{p-2}
 \widehat{v}_\lambda h d\sigma
 +\int_\Omega \partial \varphi_\lambda(\widehat{v}_\lambda)h\,dz \\
&= \int_\Omega g h \,dz  \quad \text{for all $h \in W^{1,p}(\Omega)$.}
\end{aligned} \label{eq7}
\end{equation}

In both \eqref{eq7} and \eqref{eq5} we choose 
$h=\widehat{u}_\lambda -\widehat{v}_\lambda \in W^{1,p}(\Omega)$ 
and then subtract \eqref{eq7} from \eqref{eq5}. We obtain
\begin{align*}
&\int_\Omega (|\nabla \widehat{u}_\lambda |^{p-2}\nabla \widehat{u}_\lambda
-|\nabla \widehat{v}_\lambda |^{p-2}\nabla \widehat{v}_\lambda ,
\nabla \widehat{u}_\lambda-\nabla \widehat{v}_\lambda)_{\mathbb{R}^N}\,dz \\
 &  +\widehat{\xi}\int_{\Omega}(|\widehat{u}_\lambda|^{p-2} \widehat{u}_\lambda
- |\widehat{v}_\lambda|^{p-2} \widehat{v}_\lambda) (\widehat{u}_\lambda
-\widehat{v}_\lambda) \,dz\\ 
& + \int_{\partial \Omega} \beta(z)[|\widehat{u}_\lambda|^{p-2} 
 \widehat{u}_\lambda-|\widehat{v}_\lambda|^{p-2} \widehat{v}_\lambda] 
(\widehat{u}_\lambda-\widehat{v}_\lambda) d\sigma\\ 
&   +\int_\Omega (\partial \varphi_\lambda(\widehat{u}_\lambda)
-\partial \varphi_\lambda(\widehat{v}_\lambda))(\widehat{u}_\lambda
-\widehat{v}_\lambda)\,dz = 0.
\end{align*}

Recalling that $\mathbb{R}^N \ni y \to |y|^{p-2}y$ and 
$\mathbb{R} \ni x \to \partial \varphi_\lambda(x)$ are monotone and since 
$\beta \geq 0$ (see hypothesis (H1)), we obtain
\[
 \widehat{\xi}\int_{\Omega}(|\widehat{u}_\lambda|^{p-2} \widehat{u}_\lambda
- |\widehat{v}_\lambda|^{p-2} \widehat{v}_\lambda) (\widehat{u}_\lambda
-\widehat{v}_\lambda) \,dz \leq 0
\;\Rightarrow \; \widehat{u}_\lambda =\widehat{v}_\lambda,
\]
since $\mathbb{R} \ni x \to |x|^{p-2}x$ is strictly monotone.
This proves the uniqueness of the solution of \eqref{eq4}. 
Therefore the solution map $S_\lambda: L^\infty(\Omega) \to C^1(\overline{\Omega})$ 
is well-defined. We show that $S_\lambda(\cdot)$ is continuous. To this end, 
let $g_n \to g$ in $L^\infty(\Omega)$ and set $u_n=S_\lambda(g_n)$ for all 
$n \in \mathbb{N}$ and $u=S_\lambda(g)$. We have
\begin{equation}
\begin{aligned}
&\int_\Omega |\nabla u_n |^{p-2}(\nabla u_n ,\nabla h)_{\mathbb{R}^N}\,dz
 +\widehat{\xi}\int_{\Omega} |u_n|^{p-2} u_n h \,dz \\
&+ \int_{\partial \Omega} \beta(z)|u_n|^{p-2} u_n h d\sigma
 +\int_\Omega \partial \varphi_\lambda(u_n)h\,dz \\
&= \int_\Omega g_n h \,dz  \quad \text{for all $h \in W^{1,p}(\Omega)$, all
 $n \in \mathbb{N}$.}
\end{aligned}\label{eq8}
\end{equation}

In \eqref{eq8} we choose $h=u_n \in W^{1,p}(\Omega)$. Using  (H1), 
the monotonicity of $\partial \varphi_\lambda(\cdot)$ and the fact that 
$\partial \varphi_\lambda(0)=0$, we obtain
\begin{equation}
\begin{aligned}
 & \|\nabla u_n \|_p^p + \widehat{\xi}\|u_n \|_p^p \leq c_1 \|  u _n\|
 \quad \text{for some $c_1>0$, all $n \in \mathbb{N}$},\\
&\Rightarrow \; \{u_n\}_{n \geq1} \subseteq W^{1,p}(\Omega)
\text{ is bounded}.
\end{aligned}\label{eq9}
\end{equation}
For every $n \in \mathbb{N}$, we have
\begin{equation}\label{eq10}
\begin{gathered}
- \Delta_p u_n(z) +\widehat{\xi}|u_n(z)|^{p-2}u_n(z)
+ \partial \varphi_\lambda(u_n(z)) =g_n(z) \quad  \text{for a.a. }z \in \Omega, \\
 \frac{\partial u_n}{\partial n_p} + \beta (z)|u_n|^{p-2}u_n=0 \quad \text{on  }
\partial \Omega;
\end{gathered}
\end{equation}
see \eqref{eq8} and Papageorgiou-R\v{a}dulescu \cite{Ref13}.

From \eqref{eq9}, \eqref{eq10} and  Papageorgiou-R\v{a}dulescu 
\cite[Proposition 7]{Ref14} we infer that there exists $c_2>0$ such that
$$
\|u_n\|_\infty \leq c_2 \quad \text{for all } n \in \mathbb{N}.
$$
Then  Lieberman \cite[Theorem 2]{Ref9} implies that there exist 
$\tau \in (0,1)$ and $c_3>0$ such that 
\begin{equation} 
u_n \in C^{1,\tau}(\overline{\Omega}) \quad \text{and} \quad 
\|u_n\|_{C^{1,\tau}(\overline{\Omega})}\leq c_3 \quad \text{for all }
 n \in \mathbb{N}.\label{eq11}
\end{equation}
The space $C^{1,\tau}(\overline{\Omega})$ is embedded compactly in 
$C^{1}(\overline{\Omega})$. So, from \eqref{eq11} and by passing to a 
subsequence if necessary, we may assume that
\begin{equation} \label{eq12}
u_n \to \widehat{u} \text{ in } C^{1}(\overline{\Omega}) \quad \text{as } 
n \to +\infty.
\end{equation}
Passing to the limit as $n \to +\infty$ in \eqref{eq8} and using \eqref{eq12}, 
we obtain
\begin{align*}
&\int_\Omega |\nabla \widehat{u} |^{p-2}(\nabla  \widehat{u} ,
 \nabla h)_{\mathbb{R}^N}\,dz 
 +\widehat{\xi}\int_{\Omega} | \widehat{u}|^{p-2}  \widehat{u} h \,dz\\
& + \int_{\partial \Omega} \beta(z)| \widehat{u}|^{p-2}  \widehat{u} h d\sigma 
+\int_\Omega \partial \varphi_\lambda( \widehat{u})h\,dz \\
&= \int_\Omega g h \,dz  \quad \text{for all $h \in W^{1,p}(\Omega)$,}\\ 
&\Rightarrow \;  \widehat{u} =S_\lambda(g)=u.
\end{align*}
So, for the original sequence we have that
$u_n \to u \text{ in } C^{1}(\overline{\Omega})$ which implies
$S_\lambda(\cdot)$ is continuous.

To show the compactness of $S_\lambda(\cdot)$ we need to show also that it maps 
bounded sets in $L^\infty(\Omega)$ into relatively compact sets in 
$ C^{1}(\overline{\Omega})$. So, let $B \subseteq L^\infty(\Omega)$ be bounded. 
Then as above, using  Papageorgiou-R\v{a}dulescu \cite[Proposition 7]{Ref14}, 
we have that
$$
S_\lambda(B) \subseteq L^\infty(\Omega)\text{ is bounded}.
$$
Then  Lieberman \cite[Theorem 2]{Ref9} implies that there exist $\eta \in (0,1)$ 
and $c_4>0$ such that
$$
u \in C^{1,\eta}(\overline{\Omega}) \quad \text{and} \quad 
\|u\|_{C^{1,\eta}(\overline{\Omega})}\leq c_4 \quad \text{for all }
 u \in S_\lambda(B).
$$
Exploiting the compact embedding of $C^{1,\eta}(\overline{\Omega})$ into 
$C^{1}(\overline{\Omega})$, we obtain that
\begin{align*}
&\{u : u \in S_\lambda(B)\} \subseteq C^{1}(\overline{\Omega}) \text{ is relatively 
compact},\\ 
&\Rightarrow \; S_\lambda:L^\infty(\Omega) \to C^1(\overline{\Omega}) 
\text{ is  a compact map}.
\end{align*}
\end{proof}

Let $\widehat{N}:C^1(\overline{\Omega}) \to L^\infty(\Omega)$ be defined by
$$
\widehat{N}(u)= {N}_f(u)+ \widehat{\xi}|u|^{p-2}u \quad \text{for all } 
u \in C^1(\overline{\Omega})
$$ 
with $ {N}_f(\cdot)$ being the Nemitsky map corresponding to $f$ and defined by
$$
 {N}_f(u)(\cdot)=f(\cdot,u(\cdot)) \quad \text{for all }
u \in C^1(\overline{\Omega}).
$$
We have the following result concerning this map.

\begin{proposition} \label{P5} 
If hypothesis {\rm (H2)(i)} holds, then 
$\widehat{N}:C^1(\overline{\Omega}) \to L^\infty(\Omega)$ 
is continuous and bounded (that is, maps bounded sets to bounded sets). 
\end{proposition}

\begin{proof}
It is clear from hypothesis (H2)(i), that $\widehat{N}(\cdot)$ 
is well-defined, namely it maps $C^1(\overline{\Omega})$ into $L^\infty(\Omega)$. 
Suppose that $u_n \to u$ in $C^1(\overline{\Omega})$. Then we can find
 $\rho>0$ such that
$\|u_n\|_{C^1(\overline{\Omega})}\leq \rho$.

Since $f(z,\cdot,\cdot)$ is $L^\infty$-locally H\"{o}lder continuous, 
we can find $\mu \in (0,1]$ and $c_5>0$ such that
\begin{align*}
&|f(z,u_n(z), \nabla u_n(z))- f(z,u(z), \nabla u(z))| \\ 
& \leq c_5 [|u_n(z)-u(z)|^\mu+ |\nabla u_n(z)- \nabla u(z)|^\mu]\quad 
\text{for a.a. $z \in \Omega$, all $n \in \mathbb{N}$,}\\ 
&\Rightarrow \; \widehat{N}(u_n) \to \widehat{N}(u) \quad 
 \text{in $L^\infty(\Omega)$,}\\ 
&\Rightarrow\; \widehat{N}(\cdot ) \text{ is continuous}.
\end{align*}
From (H2)(i) it is clear that $ \widehat{N}(\cdot)$ is bounded.
\end{proof}

On account of Propositions \ref{P2} and \ref{P5}, the map 
$S_\lambda \circ\widehat{N} : C^1(\overline{\Omega}) \to C^1(\overline{\Omega})$, 
$\lambda>0$, is compact. We set
$$
T_\lambda=\left\{u \in C^1(\overline{\Omega}):
 u=tS_\lambda(\widehat{N}(u)), \, 0<t<1\right\}.
$$

\begin{proposition}\label{P6}
If hypotheses {\rm (H1)--(H3)} hold, then for every $\lambda>0$, 
$T_\lambda \subseteq C^1(\overline{\Omega})$ is bounded.
\end{proposition}

\begin{proof}
Let $u \in T_\lambda$. We have
\begin{equation} \label{eq13}
\begin{aligned}
& u= t(S_\lambda \circ \widehat{N})(u),\\
&\Rightarrow \; \frac{1}{t}u=S_\lambda(\widehat{N}(u)),\\
&\Rightarrow \;  \frac{1}{t^{p-1}}\int_\Omega |\nabla u |^{p-2}
 (\nabla u, \nabla h)_{\mathbb{R}^N}\,dz
 + \frac{\widehat{\xi}}{t^{p-1}}\int_\Omega |u|^{p-2}u h \,dz \\
&\quad  + \frac{1}{t^{p-1}} \int_{\partial \Omega} \beta(z) |u|^{p-2}u h d \sigma
  +\int_\Omega \partial \varphi_\lambda\big(\frac{1}{t}u\big)h \,dz \\
&\quad =\int_\Omega [f(z,u,\nabla u)+ \widehat{\xi}|u|^{p-2}u]h\,dz \quad
 \text{for all } h \in W^{1,p}(\Omega) \\
&\Rightarrow \;  \int_\Omega |\nabla u |^{p-2}(\nabla u, \nabla h)_{\mathbb{R}^N}\,dz
 +  \widehat{\xi} \int_\Omega |u|^{p-2}u h \,dz  \\
&\quad +  \int_{\partial \Omega} \beta(z) |u|^{p-2}u h d \sigma
 +t^{p-1}\int_\Omega \partial \varphi_\lambda\big(\frac{1}{t}u\big)h \,dz \\
&\quad =t^{p-1} \int_\Omega [f(z,u,\nabla u)+ \widehat{\xi}|u|^{p-2}u]h\,dz 
 \quad  \text{for all } h \in W^{1,p}(\Omega)\,.
\end{aligned}
\end{equation}
Hypothesis (H2)(ii) implies that given $\varepsilon, \eta>0$, we can find $M>0$
 such that
\begin{equation} \label{eq14}
f(z,x,y)x \leq [\vartheta(z)+\varepsilon]|x|^p \quad
\text{for a.a. $z \in \Omega$, all $|x| \geq M$, all $ |y| \leq \eta$.}
\end{equation}
On the other hand, from hypothesis (H2)(i) we see that we can find $c_6>0$
such that
\begin{equation}\label{eq15}
f(z,x,y)x \leq c_6 (1+|y|^q) \quad \text{for a.a. $z \in \Omega$, all $|x| < M$,
all $ y \in \mathbb{R}^N$.}
\end{equation}
Since $\vartheta \in L^\infty(\Omega)$, combining \eqref{eq14} and \eqref{eq15},
we obtain
\begin{equation}\label{eq16}
f(z,x,y)x \leq [\vartheta(z)+\varepsilon]|x|^p +c_7(1+|y|^q)
\end{equation}
for a.a. $z \in \Omega$, all $x \in \mathbb{R}$, all $ y \in \mathbb{R}^N$,
some $c_7>0$.

In \eqref{eq13} we choose $h=u \in W^{1,p}(\Omega)$ and use the fact that 
$\partial \varphi_\lambda(x)x \geq 0$ for all $x \in \mathbb{R}$ 
(recall that $\partial \varphi_\lambda(\cdot)$ is monotone and
 $\partial \varphi_\lambda(0)=0$). We obtain
\begin{align*}
& \|\nabla u \|_p^p + \widehat{\xi}\|u \|_p^p 
 + \int_{\partial \Omega}\beta(z) |u|^p d \sigma\\
& \leq t^{p-1} \int_\Omega [f(z,u, \nabla u)u 
 + \widehat{\xi}|u|^p]\,dz\\ 
 & \leq t^{p-1} \int_\Omega [(\vartheta(z)+\varepsilon) |u|^p+c_7(1+|\nabla u |^q)
+ \widehat{\xi}|u|^p]\,dz \quad \text{(see \eqref{eq16})}\\ 
& \leq \int_\Omega [(\vartheta(z)+\varepsilon) |u|^p
 +c_7(1+|\nabla u |^q)+ \widehat{\xi}|u|^p]\,dz \quad 
 \text{(recall that $\widehat{\xi}>\|\vartheta\|_\infty$),}\\ 
&\Rightarrow \; \|\nabla u \|_p^p + \int_{\partial \Omega} \beta(z)|u|^p d \sigma 
 - \int_\Omega \vartheta(z)|u|^p\,dz - \varepsilon \|u\|^p 
 \leq c_8(1+\|u\|^q)  \\
&\quad \text{ for some } c_8>0,\\ 
&\Rightarrow \; (\widehat{c}-\varepsilon)\| u \|^p\leq c_8(1+\|u\|^q) 
 \quad \text{(see Lemma \ref{L3}).}
\end{align*}

By choosing  $\varepsilon \in (0,\widehat{c})$ we have that
 $$
\|u\|^p \leq c_9(1+\|u\|^q) \quad \text{for some } c_9>0.
$$
Since $q<p$ (see hypothesis (H2)(i)), we conclude that
\begin{equation}\label{eq17} 
T_\lambda \subseteq W^{1,p}(\Omega) \text{ is bounded}.
\end{equation}

For every $u \in T_\lambda$ we have
\begin{equation}\label{eq18} 
\begin{gathered}
\begin{aligned}
&- \Delta_p u(z) + \widehat{\xi}|u(z)|^{p-2}u(z)\\
&= t^{p-1}[f(z,u(z),\nabla u(z))+ \widehat{\xi}|u(z)|^{p-2}u(z)]
  -t^{p-1} \partial \varphi_\lambda\big( \frac{1}{t}u(z)\big) \\
&\quad \text{for a.a. } z \in \Omega,
\end{aligned}\\
\frac{\partial u}{\partial n_p }+ \beta(z)|u|^{p-2}u=0 \quad \text{on }
 \partial \Omega.
\end{gathered}
\end{equation}
Note that
\begin{equation}\label{eq19}
t^{p-1}\big|\partial \varphi_\lambda\big( \frac{1}{t}u(z)\big)\big| 
\leq t^{p-2}|u(z)| \leq |u(z)| \quad \text{(recall that $2 \leq p$, $0<t<1$).}
\end{equation}
Then \eqref{eq17}, \eqref{eq18}, \eqref{eq19} imply that there exists
 $c_{10}>0$ such that
$$
\|u\|_\infty \leq c_{10} \quad \text{for all $u \in T_\lambda$ (see \cite{Ref14}).}
$$
Invoking Lieberman \cite[Theorem 2]{Ref9}, we can find $s \in (0,1)$ 
and $c_{11}>0$ such that
\begin{align*}
& u \in C^{1,s}(\overline{\Omega}) \quad \text{and} \quad  
\|u\|_{C^{1,s}(\overline{\Omega})} \leq c_{11} \quad \text{for all } u \in T_\lambda,\\
& \Rightarrow \; T_\lambda \subseteq C^1(\overline{\Omega}) \quad \text{is bounded}.
\end{align*}
\end{proof}

We consider the following approximation of problem \eqref{eq1}:
\begin{equation} \label{eq20} 
 \begin{gathered} 
-\Delta_p u(z) + \partial \varphi_\lambda
(u(z))=f(z,u(z), \nabla u(z)) \quad \text{in } \Omega,\\
\frac{\partial u}{\partial n_p}+ \beta(z)|u|^{p-2}u=0 \quad
 \text{on } \partial \Omega, \; \lambda >0.
\end{gathered}
\end{equation}
On account of Propositions \ref{P5} and \ref{P6}, we can use 
Theorem \ref{T2} (the Leray-Schauder alternative principle) and have the 
following existence result for problem \ref{eq20}.

\begin{proposition}\label{P7} 
If hypotheses {(H1)--(H3)} hold, then for every $\lambda >0$ problem \ref{eq20} 
admits a solution $u_\lambda \in C^1(\overline{\Omega})$.
\end{proposition}

Finally we pass to the limit as $\lambda \to 0^+$  to produce a 
solution of the original problem \eqref{eq1}. So, we can state the following 
existence theorem for problem \eqref{eq1}.

\begin{theorem}\label{T8} 
If hypotheses {\rm (H1)--(H3)} hold, then problem \eqref{eq1} admits a 
solution $\widehat{u} \in C^1(\overline{\Omega})$.
\end{theorem}

\begin{proof}
Let $\lambda_n \to 0^+$ and let $u_n=u_{\lambda_n} \in  C^1(\overline{\Omega})$
 be a solution of problem \eqref{eq20}$_n$ (see Proposition \ref{P7}). We have
\begin{equation}
\begin{aligned}
&\int_\Omega |\nabla u_n |^{p-2}(\nabla  u_n ,\nabla h)_{\mathbb{R}^N}\,dz
+ \int_{\partial \Omega} \beta(z)| u_n|^{p-2}  u_n h d\sigma
+\int_\Omega \partial \varphi_{\lambda_n}( u_n)h\,dz\\
& = \int_\Omega f(z,u_n,\nabla u_n) h \,dz  \quad
 \text{for all $h \in W^{1,p}(\Omega)$, all $n \in \mathbb{N}$}.
\end{aligned} \label{eq21}
\end{equation}
In \eqref{eq21} we choose $h=u_n \in W^{1,p}(\Omega)$ and have
\begin{align*}
&\|\nabla u_n \|_p^p + \int_{\partial \Omega}\beta(z) |u_n|^p d \sigma \\
& \leq  \int_\Omega f(z,u_n, \nabla u_n)u_n  \,dz\\
& \leq  \int_\Omega [(\vartheta(z)+\varepsilon) |u_n|^p+c_7(1+|\nabla u_n |^q)]\,dz
 \quad \text{for all $n \in \mathbb{N}$ (see \eqref{eq16})},\\
&\Rightarrow \; \|\nabla u_n \|_p^p + \int_{\partial \Omega}
 \beta(z)|u_n|^p d \sigma - \int_\Omega \vartheta(z)|u_n|^p\,dz
  - \varepsilon \|u_n\|^p \leq c_{12}(1+\|u_n\|^q) \\
& \quad  \text{for some $c_{12}>0$, all $n \in \mathbb{N}$,}\\
&\Rightarrow \; (\widehat{c}-\varepsilon)\| u_n \|^p\leq c_{12}(1+\|u_n\|^q)
\quad \text{for all $n \in \mathbb{N}$ (see Lemma \ref{L3}).}
\end{align*}
Choose $\varepsilon \in (0,\widehat{c})$ and recall that $q<p$. We infer that
\begin{equation}\label{eq22}
\{u_n\}_{n \geq 1} \subseteq W^{1,p}(\Omega) \text{ is bounded}.
\end{equation}
So, we may assume that
\begin{equation}\label{eq23}
u_n \xrightarrow{w} 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}

Since $\partial \varphi_{\lambda_n}(\cdot)$ is Lipschitz continuous, 
from Marcus-Mizel \cite{Ref10} (see also Gasi\'nski-Papageorgiou 
\cite[Proposition 2.4.24, p. 194]{Ref3}, we have
$$
\partial \varphi_{\lambda_n}(u_n(\cdot)) \in W^{1,p}(\Omega) \quad \text{for all } 
n \in \mathbb{N}.
$$
Hence we can choose $h= \partial \varphi_{\lambda_n}(u_n) \in W^{1,p}(\Omega)$ 
in \eqref{eq21}. We  obtain
\begin{equation}
\begin{aligned}
&\int_\Omega |\nabla u_n |^{p-2}(\nabla  u_n ,
 \nabla (\partial \varphi_{\lambda_n}(u_n)))_{\mathbb{R}^N}\,dz\\
&  + \int_{\partial \Omega} \beta(z)| u_n|^{p-2}  u_n \partial
 \varphi_{\lambda_n}(u_n) d\sigma +\| \partial \varphi_{\lambda_n}( u_n)\|_2^2\\
& = \int_\Omega f(z,u_n,\nabla u_n) \partial \varphi_{\lambda_n}(u_n) \,dz  \quad
\text{for all $n \in \mathbb{N}$}.
\end{aligned} \label{eq24}
\end{equation}
From the chain rule for Sobolev functions of Marcus-Mizel \cite{Ref10}
(see also Gasi\'nski-Papageorgiou \cite[p. 194]{Ref3}), we have
\begin{equation*} %\label{eq25}
\nabla (\partial \varphi_{\lambda_n}(u_n))=(\partial \varphi_{\lambda_n})'(u_n)
\nabla u_n \quad \text{for all $n \in \mathbb{N}$}.
\end{equation*}
Hence
\begin{equation}\label{eq26}
\int_\Omega |\nabla u_n |^{p-2}(\nabla  u_n ,\nabla
 (\partial \varphi_{\lambda_n}(u_n)))_{\mathbb{R}^N}\,dz
= \int_\Omega|\nabla u_n |^{p}(\partial \varphi_{\lambda_n})'(u_n) \,dz\quad
\end{equation}
for all $n \in \mathbb{N}$.
Since $\partial \varphi_{\lambda_n}(\cdot)
$ is nondecreasing, we have $(\partial \varphi_{\lambda_n})'(u_n) \geq 0$.
Therefore
\begin{equation} \label{eq27}
 0 \leq \int_\Omega |\nabla u_n |^{p-2}(\nabla  u_n ,
\nabla (\partial \varphi_{\lambda_n}(u_n)))_{\mathbb{R}^N}\,dz \quad
\text{for all $n \in \mathbb{N}$ (see \eqref{eq26})}.
\end{equation}

Recall that $\partial \varphi_{\lambda_n}(u_n)u_n \geq 0$ for all
$n \in \mathbb{N}$. So
\begin{equation}\label{eq28}
0 \leq \int_{\partial \Omega} \beta(z)| u_n|^{p-2} 
 u_n \partial \varphi_{\lambda_n}(u_n) d\sigma \quad 
\text{for all $n \in \mathbb{N}$ (see hypothesis (H1))}.
\end{equation}
We return to \eqref{eq24} and use \eqref{eq27} and \eqref{eq28}. We obtain
\begin{equation} \label{eq29}
 \| \partial \varphi_{\lambda_n}( u_n)\|_2^2    
\leq \int_\Omega f(z,u_n,\nabla u_n) \partial \varphi_{\lambda_n}(u_n) \,dz  
\quad \text{for all $n \in \mathbb{N}$}.
\end{equation}
From hypothesis (H2)(i) we have
\begin{equation}
\begin{aligned}
&|f(z,u_n(z), \nabla u_n(z))| \leq a(z) [1+|u_n(z)|^{p-1}+ |\nabla u_n(z)|^{q}]
 \quad\text{for a.a. $z \in \Omega$,}\\
&\Rightarrow \; |f(z,u_n(z), \nabla u_n(z))|^2
 \leq c_{13} [1+|u_n(z)|^{2(p-1)}+ |\nabla u_n(z)|^{2q}] \\
& \quad  \text{for some $c_{13}>0$, a.a. $z \in \Omega$, all $n \in \mathbb{N}$,}\\
&\Rightarrow \;  N_f(u_n) \in  L^2(\Omega) \text{ and $\| N_f(u_n)\|_2 \leq c_{14}$
for some $c_{14}>0$, all $n \in \mathbb{N}$};
\end{aligned}\label{eq30}
\end{equation}
see \eqref{eq22} and recall that $2(p-1) \leq p^*$,  $2q=\max \{p-1,2\}$.
Then from \eqref{eq29} and the Cauchy-Schwarz inequality, we have
\begin{equation} \label{eq31}
\begin{aligned}
 & \| \partial \varphi_{\lambda_n}( u_n)\|_2^2
\leq c_{14} \| \partial \varphi_{\lambda_n}( u_n)\|_2
\quad \text{for all $n \in \mathbb{N}$ (see \eqref{eq30})}  \\
& \Rightarrow \; \{\partial \varphi_{\lambda_n}( u_n) \}_{n \geq 1}
\subseteq L^2(\Omega) \text{ is bounded}.
\end{aligned}
\end{equation}
We may assume that
\begin{equation}\label{eq32}
\partial \varphi_{\lambda_n}( u_n) \xrightarrow{w} e \text{ in } L^2(\Omega)
\quad \text{as } n \to + \infty.
\end{equation}

In \eqref{eq21} we choose $h=u_n-u \in W^{1,p}(\Omega)$, pass to the limit as 
$n \to + \infty$ and use \eqref{eq23}, \eqref{eq30}, \eqref{eq32}. Then
\begin{equation}  \label{eq33}
  \lim_{n \to +\infty} \int_\Omega |\nabla u_n|^{p-2}(\nabla u_n,
\nabla (u_n-u))_{\mathbb{R}^N}\,dz=0\; \Rightarrow \;
 u_n \to u \text{ in } W^{1,p}(\Omega) 
\end{equation}
as $n \to + \infty$;
see Motreanu-Motreanu-Papageorgiou \cite[Proposition 2.72, p. 40]{Ref11}.
We have
\begin{equation}\label{eq34}
 N_f(u_n) \to N_f(u) \text{ in } L^{p'}(\Omega) \quad \text{(see hypothesis (H2)(i))}.
\end{equation}

Passing to the limit as $n \to + \infty$ in \eqref{eq21} and using 
\eqref{eq32}, \eqref{eq33}, \eqref{eq34}, we obtain
\begin{equation} \label{eq35} 
\begin{aligned}
&\int_\Omega |\nabla u |^{p-2}(\nabla u, \nabla h)_{\mathbb{R}^N}\,dz 
+ \int_{\partial \Omega}\beta(z) |u|^{p-2}u h d \sigma
 + \int_\Omega e h \,dz \\
&= \int_\Omega f(z,u, \nabla u) h \,dz
\end{aligned}
\end{equation}
for all $h \in W^{1,p}(\Omega)$. Also we have
\begin{equation*}% \label{eq36}
 \partial \varphi_{\lambda_n}(u_n(z)) \in \partial \varphi( J_{\lambda_n}(u_n(z)) 
\quad \text{for a.a. $z \in \Omega$, all $n \in \mathbb{N}$,}
\end{equation*}
with $J_{\lambda_n}=({\rm id } - \lambda_n \partial \varphi)^{-1}$ 
being the resolvent map. So, we have
\begin{equation}\label{eq37}
J_{\lambda_n}(u_n(z)) + \lambda_n \partial \varphi_{\lambda_n}(u_n(z))
=u_n(z)\quad \text{for a.a. $z \in \Omega$, all $n \in \mathbb{N}$.}
\end{equation}
If we set $\widehat{J}_{\lambda_n}(u_n(\cdot))=
J_{\lambda_n}(u_n(\cdot)) \in L^2(\Omega)$ for all $n \in \mathbb{N}$  
(see Gasi\'nski-Papageorgiou \cite[p. 323]{Ref3}), from \eqref{eq31}, \eqref{eq33}
 and \eqref{eq37} we have
\begin{equation} \label{eq38}
\begin{aligned}
& \widehat{J}_{\lambda_n}(u_n) -u_n \to 0 \text{ in } L^2(\Omega)
 \quad \text{(recall that $\lambda_n \to 0^+$),}\\
&\Rightarrow \; \widehat{J}_{\lambda_n}(u_n)  \to  u \text{ in } L^2(\Omega),\\
&\Rightarrow \; e(z) \in \partial \varphi(u(z))\quad \text{for a.a }z \in \Omega,
\end{aligned}
\end{equation}
see \eqref{eq32} and use Gasi\'nski-Papageorgiou
\cite[Proposition 3.2.15, p. 308]{Ref3}). Then from \eqref{eq35} and \eqref{eq38}
we infer that $u \in W^{1,p}(\Omega)$ is a solution of \eqref{eq1}.
As before the nonlinear regularity theory of Lieberman \cite{Ref9} implies that
$u \in C^1(\overline{\Omega})$.
\end{proof}

\section{Example}\label{Sec4}

In this section we see a particular case of problem \eqref{eq1}, 
which corresponds to a variational inequality.
Let
$$ 
\varphi(x)=i_+(x)=\begin{cases} 0 & \text{if } x \geq 0,\\
 + \infty & \text{if } x <0,
\end{cases}
$$
(the indicator function of the positive semiaxis $\mathbb{R}_+=[0,+\infty)$).
Then $\varphi \in \Gamma_0(\mathbb{R})$ and we have
$$
\partial \varphi(x)=N_{\mathbb{R}_+}(x)\quad \text{(the normal cone to $\mathbb{R}_+$ 
at $x$)};
$$
see Gasi\'nski-Papageorgiou \cite[p. 526]{Ref3}. We have
$$
N_{\mathbb{R}_+}(x)=\{x^* \in \mathbb{R}: x^*(c-x) \leq 0 \text{ for all }c \geq 0\}.
$$
Evidently $0 \in \partial \varphi(0)$. We consider a reaction term $f(z,x,y)$ 
satisfying  (H2). Then according to Theorem \ref{T8} we can find
$$
u \in C^1(\overline{\Omega}) \text{ with } u(z) \geq 0 \quad \text{for all }
 z \in \overline{\Omega}
$$
which satisfies
\begin{gather*}
 - \Delta_p u(z)=f(z,u(z), \nabla u(z)) \quad \text{for a.a. } 
z \in \Omega_+ =\{z \in \Omega : u(z)>0\},\\ 
 - \Delta_p u(z) \geq f(z,u(z), \nabla u(z)) \quad \text{for a.a. } 
 z \in \Omega_0 =\{z \in \Omega : u(z)=0\},\\
 (- \Delta_p u(z))u(z)=f(z,u(z), \nabla u(z))u(z) \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*}

\subsection*{Acknowledgments}
The authors wish to thank the anonymous referee for his/her helpful remarks.


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