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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 309, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/309\hfil Existence of solutions]
{Existence of solutions for a non-variational system of elliptic PDE's
via topological methods}

\author[F. Soltani, H. Yazidi \hfil EJDE-2016/309\hfilneg]
{Fethi Soltani, Habib Yazidi}

\address{Fethi Soltani \newline
Department of Mathematics,
Faculty of Science,
Jazan University,
P.O. Box 277, Jazan 45142, Saudi Arabia}
\email{fethisoltani10@yahoo.com}

\address{Habib Yazidi (corresponding author)\newline
University of Tunis,
National School of Engineering of Tunis,
Depatment of Mathematics, 5 Street Taha Hssine,
Bab Mnara 1008 Tunis, Tunisia}
\email{habib.yazidi@gmail.com}

\thanks{Submitted April 26, 2016. Published November 30, 2016.}
\subjclass[2010]{35B45, 35P30, 35J65, 35J55}
\keywords{Biharmonic equation; non-variational system;  a priori estimates}

\begin{abstract}
 In this article, we prove the existence of solutions for a non-variational
 system of elliptic PDE's. Also we study a system of bi-Laplacian equations
 with two nonlinearities and without variational assumptions.
 First, we prove a priori solution estimates, and then we use
 fixed point theory, to deduce the existence  of solutions.
 Finally, to complement of the existence theorem, we establish a
 non-existence result.
\end{abstract}

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

\section{Introduction}

We consider the  problem
\begin{equation}
\begin{gathered}
\Delta^2u=g(v),\quad v>0 \quad \text{in }B,\\
\Delta^2v=f(u),\quad u>0 \quad \text{in } B,\\
u=0,\quad \frac{\partial u}{\partial \nu}=0,\quad v=0,\quad
\frac{\partial v}{\partial \nu}=0 \quad \text{on }\partial B,
\end{gathered} \label{eq1}
\end{equation}
where $B$ is the unit ball in $\mathbb{R}^{N}$ ($N> 4$), the functions
$f$ and $g$ are continuous and positive on
$(0,\infty)$ satisfying $f(0)=0$ and $g(0)=0$.

Existence results for elliptic nonlinear systems have earned a lot of
interest in recent years, especially when the nonlinear term appears
as a source in the equation, supplemented by the boundary conditions
of Dirichlet or Neumann. There are two broad classes of systems,
the first one with a variational structure, namely Hamiltonian or gradients systems;
see \cite{1,12,121}.
The second one is the class of non-variational problems, which can be treated
via topological arguments. For this type of results see \cite{2,5,6,7}.

In this work, we address the existence problem for a given system without
a variational assumption. More precisely we consider the non-variational case
of $f$ and $g$. Using a topological method (a fixed-point argument), some
sufficient conditions for the study of this problem were established.
It was shown that a priori estimate of positive solutions for elliptic
equations provides a great deal of information about the existence and the
structure set of positive solutions \cite{4}, \cite{6}, \cite{13} and \cite{14}.
Our objective is to prove existence of results and a priori estimate of solutions.

This type of question was addressed in several works dealing with the Laplacian
problem. One of the pioneering studies in this direction was \cite{15}.
In this work, we consider the bi-Laplacian operator. In recent years several
 authors studied polyharmonic problems and a lot of interesting and significant
results were obtained see \cite{71}, \cite{11}, \cite{141} the references therein,
see also  \cite{161} for more general operator, namely nonhomogeneous
differential operator with variable exponents.

The rest of the article is divided into two sections.
In Section 2, we establish preliminary results which are helpful to study
the bi-Laplacian system \eqref{eq1}. Section 3 was devoted to present and
prove the main results of this work.

\section{Preliminary Results}

Consider the problem \eqref{eq1} for radially symmetric solutions, let $r=|x|$,
$u=u(r)$ and $v=v(r)$,
\begin{equation}
\begin{gathered}
u^{(4)}+\frac{2(N-1)}{r}u^{(3)}+\frac{(N-1)(N-3)}{r^2} u''
 -\frac{(N-1)(N-3)}{r^{3}} u'=g(v),\\
v>0\quad \text{for } r \in (0, 1),\\
v^{(4)}+\frac{2(N-1)}{r}v^{(3)}+\frac{(N-1)(N-3)}{r^2} v''
-\frac{(N-1)(N-3)}{r^{3}} v'=f(u),\\
u>0 \quad \text{for }r \in (0, 1),\\
u'(0)=0=v'(0),\quad u^{(3)}(0)=0=v^{(3)}(0),\\
  u(1)=0=v(1),\quad
 u'(1)=0=v'(1).
\end{gathered} \label{eqp1}
\end{equation}
We remark that any solution $(u(r), v(r))\in \left(C^{4}(0,1)\right)^2$ of
\eqref{eqp1} is a radial symmetric solution of
\eqref{eq1}.

We recall the following result from \cite[Lemma 2]{17}, which gives
more information concerning the eigenvalue problem for the
operator $\Delta^2$.

\begin{lemma} \label{lem2.1}
There is a $\mu_1> 0$ such that the problem
$$
\Delta^2v=\mu_1 v \quad \text{in } B,\quad
v=\frac{\partial v}{\partial \nu}\quad \text{on }\partial B
$$
possesses a positive, radial symmetric solution $\varphi_1(x)$
 which satisfies, for some positive constants $C_1$ and $C_2$,
\begin{equation}
C_1 (1-|x|)^2\leq \varphi_1(x)\leq C_2 (1-|x|)^2,\quad x\in \overline{B}.
\label{eqvalue}
\end{equation}
\label{lm1}
\end{lemma}


We recall from \cite{17}, see also \cite{11}, that the Green function
$G(r,s)$ for the linear problem corresponding to \eqref{eqp1} is defined,
for $N> 4$, by
\begin{equation}
G(r, s)=\begin{cases}
a_{N}(s)+r^2 b_{N}(s), &\text{for } 0\leq r\leq s\leq 1\\
(\frac{s}{r})^{N-1}(a_{N}(r)+s^2 b_{N}(r)),\ &\text{for }
0\leq s\leq r\leq 1,
\end{cases} \label{eqp2}
\end{equation}
where
$$
a_{N}(t)=\frac{t^{3}}{4(N-2)(N-4)}[2+(N-4)t^{N-2}-(N-2)t^{N-4}]
$$
and
$$
b_{N}(t)=\frac{t}{4N(N-2)}[N t^{N-2}-(N-2)t^{N}-2].
$$
The kernel $G(r, s)$ has the following properties (see \cite{17}). There
exists a positive constant $C$ such that
\begin{gather}
0\leq G(r,s)\leq C s^{N-1}(1-s)^2\big(\max(r,s)\big)^{4-N},\label{eqg2} \\
\frac{\partial }{\partial r}G(r,s)(r,s)\leq 0, \label{eqg1} \\
\frac{\partial ^2}{\partial r^2}G(r,s)\big{|}_{r=1}
 =\frac{1}{2}s^{N-1}(1-s^2).\label{eqg3}
\end{gather}
Therefore,  problem \eqref{eqp1} is transformed into
\begin{equation}
\begin{gathered}
u(r)=\int_{0}^{1}G(r, s)g(v(s))\mathrm{d}s,\\
v(r)=\int_{0}^{1}G(r, s)f(u(s))\mathrm{d}s.
\end{gathered} \label{eqp3}
\end{equation}
It is well known that problem \eqref{eqp1} and problem \eqref{eqp3}
are equivalent.

In the study of the problem \eqref{eq1}, we need
the following eigenvalue problem
\begin{equation}
\begin{gathered}
\Delta ^2\phi=\lambda_2 \psi \quad \text{in } B,\\
\Delta ^2\psi=\lambda_1 \phi \quad \text{in } B,\\
\phi=0=\frac{\partial\phi}{\partial \nu},\quad
\psi=0=\frac{\partial\psi}{\partial \nu} \quad \text{on }
\partial B,
\end{gathered} \label{eqp4}
\end{equation}
where $\lambda_1, \lambda_2>0$.

Let $\varphi_1$ be the corresponding eigenfunction of $\mu_1$
which is the first eigenvalue of $\Delta^2$ on the unit ball $B$,
we have the following result.

\begin{lemma} \label{lm2}
Assume that $\lambda_1 \lambda_2=\mu_1^2$, then the problem \eqref{eqp4}
has a positive solution $(\phi, \psi)$ satisfying (modulo a constant)
 $\phi=\frac{1}{\sqrt{\lambda_1}} \varphi_1$ and
$\psi=\frac{1}{\sqrt{\lambda_2}} \varphi_1$.
\end{lemma}

\begin{proof}
According to the idea developed in \cite{18} for a Laplacian
eigenvalue problem, we define
\begin{gather}
w_1=\sqrt{\lambda_1} \phi, \label{eqlm1} \\
w_2=\sqrt{\lambda_2} \psi. \label{eqlm2}
\end{gather}
We replace \eqref{eqlm1} and \eqref{eqlm2} in the problem
\eqref{eqp4}, we obtain
\begin{equation}
\begin{gathered}
\Delta ^2 w_1=\sqrt{\lambda_1\lambda_2} w_2 \quad\text{in } B,\\
\Delta ^2 w_2=\sqrt{\lambda_1\lambda_2 } w_1 \quad\text{in } B,\\
w_1=0=\frac{\partial w_1}{\partial \nu},\quad
 w_2=0=\frac{\partial w_2}{\partial \nu} \quad \text{on }\partial B.
\end{gathered} \label{eqlm3}
\end{equation}
Adding the two first equations of \eqref{eqlm3}, we write
\begin{equation}
\begin{gathered}
\Delta ^2 (w_1+w_2)=\sqrt{\lambda_1\lambda_2} (w_1+w_2) \quad\text{in } B,\\
w_1+w_2=0,\,\, \frac{\partial(w_1+w_2)}{\partial \nu}=0 \quad\text{on } \partial B.
\end{gathered} \label{eqlm4}
\end{equation}
Subtracting the two first equations of \eqref{eqlm3}, we write
\begin{equation}
\begin{gathered}
\Delta ^2 (w_1-w_2)=\sqrt{\lambda_1\lambda_2} (w_2-w_1) \quad\text{in } B,\\
w_1-w_2=0,\quad \frac{\partial (w_1-w_2)}{\partial \nu}=0 \quad \text{on }
\partial B.
\end{gathered} \label{eqlm5}
\end{equation}
We multiply \eqref{eqlm5} by $w_1-w_2$ and we make a two
integration by parts, we obtain
\begin{equation*}
\int_{B}|\Delta(w_1-w_2)|^2\mathrm{d}x
=-\sqrt{\lambda_1\lambda_2} \int_{B}|w_1-w_2|^2\mathrm{d}x,
\end{equation*}
which proves that $w_1=w_2$ in $\overline{B}$. Since
$\sqrt{\lambda_1 \lambda_2}=\mu_1$ and looking at the properties of the
eigenvalue problem for the bi-Laplacian, we have that the problem \eqref{eqlm4}
has the first eigenfunction $\varphi_1$ as the only solution.
Then, for a positive constant $C$, we have
$w_1=w_2=C \varphi_1$ therefore $\phi=C \frac{1}{\sqrt{\lambda_1}} \varphi_1$
and $\psi=C \frac{1}{\sqrt{\lambda_2}} \varphi_1$.
\end{proof}

We end this section by giving the following identity which plays an
important role in the study of our problem.
Let $F$ and $G$ be, respectively, the primitives of $f$ and $g$ such that
 $F(0)=0$ and $G(0)=0$.

\begin{lemma} \label{pohozaev}
Let $(u, v)$ a solution of the system \eqref{eq1} and $\alpha$,
$\beta$ are some positive constants. We have the  identity
\begin{equation}
\begin{aligned}
\int_{\partial B}(\Delta u, \Delta v) (x. \nu)
\mathrm{d}\sigma_{x}
&= \int_{B}\left(N F(u)+N G(v)-\alpha u f(u)-\beta v g(v) \right)
\mathrm{d}x \\
&\quad +(N-4-(\alpha+\beta))\int_{B}(\Delta u, \Delta v) \mathrm{d}x.
\end{aligned} \label{idserrin}
\end{equation}
\end{lemma}

\begin{proof}
According to  \cite[Proposition 4]{15}, \cite[Theorem 2.1]{18} and
by easy computation, the following identity holds
\begin{equation}
\begin{aligned}
&\frac{\partial}{\partial x_{i}}\Big[x_{i} L
 -\Big(x_{k}\frac{\partial u_{l}}{\partial x_{k}}+a_{l} u_{l} \Big)
 \Big(L-{p_{i}}-\frac{\partial}{\partial x_{j}}L_{r_{ij}} \Big) \\
& -\frac{\partial}{\partial x_{j}}
 \Big(x_{k}\frac{\partial u_{l}}{\partial x_{k}}+a_{l}u_{l} \Big)L_{r_{ij}}\Big]\\
&= N L+x_{i} L_{x_{i}}-a_{l} u_{l} L_{u_{l}}
 -(a_{l}+1)\frac{\partial u_{l}}{\partial x_{i}} L_{p_{i}}
 -(a_{l}+2)\frac{\partial ^2u_{l}}{\partial x_{i}\partial x_{j}} L_{r_{ij}},
\end{aligned} \label{idserrin1}
\end{equation}
where $L=L(x, U, p, r)$ is a lagrangian with $U=(u_1, u_2)$, $p=(p^{k}_{i})$,
$p^{k}_{i}=\frac{\partial u_{k}}{\partial x_{i}}$, $r=(r_{ij})$, $i=1,\dots,N$
 and $a_1$, $a_2$ are some constants. We apply the last identity to the
Lagrangian associate with  problem \eqref{eq1};
$L=L(x, U, \nabla U, \Delta U)=(\Delta u, \Delta v )+ F(u)+G(v)$,
$a_1=\alpha$, $a_2=\beta$.
We integrate \eqref{idserrin1} over $B$ and use the condition $u=0=v$,
$\frac{\partial u}{\partial \nu}=0=\frac{\partial v}{\partial \nu}$ on
 $\partial B$,  we obtain \eqref{idserrin}.
\end{proof}

\begin{remark} \label{rmk2.1} \rm
Looking at \eqref{idserrin}, if we take $\alpha+\beta=N-4$, we see
that the critical conditions on $f$ and $g$ are $N F(u)-\alpha u f(u)=0$
and $N G(v)-(N-4-\alpha) v g(v)=0$, then
\begin{equation*}
\frac{f(u)}{F(u)}=\frac{N/ \alpha}{u} \quad \text{and}\quad
\frac{g(v)}{G(v)}=\frac{N/(N-4-\alpha)}{v}.
\end{equation*}
An easy computation gives, for some positive constants $c_1$ and $c_2$,
\begin{equation*}
f(u)= c_1 u^{\frac{N}{\alpha}-1}\quad \text{and}\quad
g(v)=c_2 v^{\frac{N}{N-4-\alpha}-1}.
\end{equation*}
\end{remark}

\section{Main results and their proofs}

Let $F$ and $G$ be, respectively, the primitives of $f$ and $g$ such that
$F(0)=0$ and $G(0)=0$.
We introduce the following critical exponents associate to the
 system \eqref{eq1} by
\begin{equation}
p^{\star}=\frac{N-\alpha}{\alpha},\quad q^{\star}=\frac{4+\alpha}{N-4-\alpha},\quad
\text{where $\alpha\in ((N-4)/2,\,N/2)$}.
\label{exposant}
\end{equation}
Easily, we see that
\[
\frac{1}{p^{*}+1}+\frac{1}{q^{*}+1}=\frac{N-4}{N}.
\]
Our main results are the following.

\begin{theorem} \label{th1}
Suppose that $f$ and $g$ satisfy the following hypothesis
\begin{itemize}
\item[(i)] $\liminf_{s\to \infty}f(s)s^{-1}> \lambda_1$ and
 $\limsup_{s\to 0}f(s)s^{-1}< \lambda_1$,\\
 $\liminf_{s\to \infty}g(s)s^{-1}> \lambda_2$ and
 $\limsup_{s\to 0}g(s)s^{-1}< \lambda_2$.

\item[(ii)] $N F(s)-\alpha s f(s)\geq \theta_1 sf(s)$, $s> 0$, for some
$\theta_1\geq 0$, \
$N G(s)-\beta s g(s)\geq \theta_2sg(s)$, $s> 0$, for some $\theta_2\geq 0$,
and $\alpha$ and $\beta$ are positive reals satisfying $\alpha+\beta=N-4$.
\end{itemize}
We suppose that
\begin{itemize}
\item[(H1)] there exists a constant $C>0$ such that for every positive
solution $(u, v)$ of problem~\eqref{eq1} verifies
$\|u\|_{\infty}\leq C$ and $\|v\|_{\infty}\leq C$.
\end{itemize}
Then the system \eqref{eq1} has a positive solution.
\end{theorem}

\begin{theorem} \label{th2}
Under conditions (i) and (ii) on $f$ and $g$, the condition {\rm (H1)} is
satisfied, namely, every positive solution of system \eqref{eq1} is bounded in
 $L^{\infty}$.
\end{theorem}

The non-existence result is the following.

\begin{theorem} \label{th3}
Suppose that $f$ and $g$ satisfy
\begin{equation*}
N F(t)-\alpha t f(t)\leq 0\quad \text{and} \quad
N G(t)-\beta t\, g(t)\leq 0 \quad \text{for } t>0.
\end{equation*}
Then  problem \eqref{eq1} has no nontrivial solution
$(u,\,v) \in \big(C^{4}(\bar{B})\big)^2$.
\end{theorem}

\begin{remark} \label{rm1} \rm
Under  conditions (i) and (ii) on $f$ and $g$ of Theorem \ref{th1} we have
$$
\lim_{t\to \infty}\frac{f(t)}{t^{p^{*}}}=0 \quad\text{and}\quad
\lim_{t\to \infty}\frac{g(t)}{t^{q^{*}}}=0.
$$
\end{remark}

Indeed, from condition (i), we see that there exists $t_{0}>0$ such that
 $f(t)>0$ and $g(t)>0$ for $t> t_{0}$. Hence, for $t>t_{0}$, from condition
(ii) we write
\begin{equation}
N F(t)\geq -\theta_1+\eta t f(t) \quad \text{and}\quad
N G(t)\geq -\theta_2+\mu t g(t), \label{eqrm1}
\end{equation}
where $\eta=\alpha+\theta_1$ and $\mu=\beta+\theta_2$.
Therefore,
\begin{equation*}
F'(t)-\frac{N}{\eta t} F(t)\leq \frac{\theta_1}{\eta t} \quad \text{and}\quad
G'(t)-\frac{N}{\mu t} G(t)\leq \frac{\theta_{3}}{\mu t}.
\end{equation*}
Multiplying the two last inequalities
 by $t^{-\frac{N}{\eta}}$ and $t^{-\frac{N}{\mu}}$, respectively, we obtain
\begin{equation*}
\frac{\mathrm{d}}{\mathrm{d}t}\Big(t^{-\frac{N}{\eta}} F(t)
\Big)\leq \frac{\theta_1}{\eta}t^{-1-\frac{N}{\eta}} \quad
\text{and}\quad
\frac{\mathrm{d}}{\mathrm{d}t}\Big(t^{-\frac{N}{\mu}} G(t)
\Big)\leq \frac{\theta_1}{\eta}t^{-1-\frac{N}{\mu}}.
\end{equation*}
We deduce, for some positive constants $C_1$ and $C_2$, that
\begin{equation*}
F(t)\leq C_1  t^{N/\eta} \quad \text{and}\quad G(t)\leq C_2 t^{N/\mu}.
\end{equation*}
We replace into \eqref{eqrm1}, we obtain, for $t$ large enough, that for some
 positive constant $C$ and $\bar{C}$,
\begin{equation*}
f(t) \leq C t^{\frac{N}{\eta}-1}\quad \text{and}\quad
g(t) \leq \bar{C} t^{\frac{N}{\mu}-1} .
\end{equation*}
Or $\alpha+\beta=N-4$, then, since $\eta=\alpha+\theta_1$,
$\mu=\beta+\theta_2$ and $\theta_1$, $\theta_2> 0$, we have $\eta+\mu> N-4$.

Now, we return to the proofs of Theorem \ref{th1} and
Theorem \ref{th2}. The proof of Theorem \ref{th1} is based on
a topological argument. More precisely, we apply the following
fixed point theorem \cite{9}, see also \cite{16}.

\begin{theorem}[\cite{9}] \label{thfixed}
Let $C$ be a cone in a Banach space $X$ and  $\Phi:C\to C$ a compact map such
 that $\Phi(0)=0$. Assume that there exist numbers $0<r<R$ such that
\begin{itemize}
\item[(a)] $x\neq \lambda \Phi(x)$ for $0\leq \lambda \leq 1$ and $\|x\|=r$,
\item[(b)] there exists a compact map $F:\overline{B_{R}}\times [0,\,\infty)\to C$
such that
\begin{gather*}
F(x,0)=\Phi(x) \quad \text{if }\|x\|=R,\\
F(x,\mu)\neq x \quad \text{if $\|x\|=R$ and $0\leq \mu< \infty$,}\\
F(x,\mu)\neq x \quad \text{if $x\in \overline{B_{R}}$ and $ \mu \geq \mu_{0}$.}
\end{gather*}
\end{itemize}
Then if $U=\{x\in C:\,r<\|x\|<R\}$ and
$B_{\rho}= \{x\in C:\|x\|< \rho\}$, we
have
$$
i_{C}(\Phi,\,B_{R})=0,\quad i_{C}(\Phi,\,B_{r})=1, \quad i_{C}(\Phi,\,U)=-1,
$$
where $i_{C}(\Phi,\Omega)$ denotes the index of $\Phi$ with respect to $\Omega$.
In particular, $\Phi$ has a fixed point in $U$.
\end{theorem}

\begin{proof}[Proof of Theorem \ref{th1}]
We apply Theorem \ref{thfixed}, then consider the Banach space
$X=\big(C^{*}(0, 1)\big)^2$, where $C^{*}(0, 1)$ denote
the space of continuous bounded functions defined on $(0, 1)$,
endowed with the norm $ \|u\|:=\sup_{t \in (0,1)}|u(t)|$.
We define the cone $C$ by
$$
C:=\big\{w\in X: w(t)\geq 0, \text{ for all $t\in(0,1)$} \big\},
$$
where $w=(y, z)\geq 0$ means that $y\geq 0$
and $z\geq 0$. We define the compact map $\Phi: X\to X$ by
$$
\Phi(w)(r)=\int_{0}^{1}G(r,s)h(w(s))\,\mathrm{d}s,\quad
 h(w)=(g(v), f(u)).
$$
We observe that a fixed point of $\Phi$ is a solution of
\eqref{eqp3} and then a solution of \eqref{eq1}.

Now we shall verify the conditions of Theorem \ref{thfixed}.
\smallskip

\noindent\textbf{Verification of condition (a):}
From condition (i) of Theorem \ref{th1} there exists positive constants
$q_1<1$ and $q_2< 1$ such that $f(u(x))\leq q_1 \lambda_1u(x)$ and
$g(v(x))\leq q_2 \lambda_2 v(x)$. Then we have
\begin{align*}
 \lambda_2\int v \psi \mathrm{d}x
&= \int v\Delta^2 \phi \mathrm{d}x
 =\int \Delta^2 v \phi \mathrm{d}x\\
&=\int f(u) \phi \mathrm{d}x\leq q_1\lambda_1 \int u \phi \mathrm{d}x.
\end{align*}
On the other hand,
\begin{align*}
 \lambda_1\int u \phi \mathrm{d}x
&= \int u \Delta^2 \psi \mathrm{d}x
 =\int \Delta^2 u \psi \mathrm{d}x\\
&= \int g(v) \psi \mathrm{d}x
\leq  q_2\lambda_2 \int v \psi \mathrm{d}x.
\end{align*}
Combining these two inequalities, we write
\begin{gather}
 \lambda_2\int v \psi \mathrm{d}x\leq
q_1\lambda_1 \int u \phi \mathrm{d}x\leq q_1q_2
\lambda_2\int v \psi \mathrm{d}x, \label{eqm1} \\
 \lambda_1\int u \phi \mathrm{d}x\leq
q_2\lambda_2 \int v \psi \mathrm{d}x\leq q_2q_1
\lambda_1\int u \phi \mathrm{d}x. \label{eqm2}
\end{gather}
Or $q_1 q_2< 1$ then \eqref{eqm1} and \eqref{eqm2} give a contraction
since the integrals are nonzero. Moreover, if $u$ and $v$ are replaced
respectively by $\lambda\,u$ and $\lambda\,v$ in, respectively, \eqref{eqm1}
and \eqref{eqm2}, for $\lambda \in [0, 1]$, then a contradiction also
follows and therefore
$$
w(t)\neq \lambda \Phi(w(t))\quad \text{with}\quad
\lambda \in [0, 1],\quad \|w\|=r,\quad w \in C.
$$

\noindent\textbf{Verification of condition (b):}
Define the compact mapping $F:C\times [0, \infty)\to C$ by
\begin{equation}
F(w, \mu)(r)=\Phi(w+\mu)(r) \label{eqm3}
\end{equation}
Easily, we see that $F(w, 0)=\Phi(w)$. The condition (i) of
Theorem \ref{th1} gives the existence of constants
$k_1>\lambda_1$, $k_2> \lambda_2$ and $\mu_{0}>0$ such
that $f(y+\mu)\geq k_1 y$ and $g(z+\mu)\geq k_2 z$ if
 $\mu\geq \mu_{0}$ for all $(y, z)\geq (0, 0)$. We have
\begin{align*}
\lambda_2\int v \psi \mathrm{d}x
&= \int v \Delta^2\phi \mathrm{d}x
 =\int \Delta^2v \phi \mathrm{d}x \\
&= \int f(u)\phi \mathrm{d}x\geq k_1 \int u \phi \mathrm{d}x\geq
\lambda_1 \int u \phi \mathrm{d}x.
\end{align*}
Or
\begin{align*}
\lambda_1\int u \phi \mathrm{d}x
&= \int u \Delta^2\psi \mathrm{d}x=\int \Delta^2u \psi \mathrm{d}x \\
&= \int g(v)\psi \mathrm{d}x\geq k_2\int v \psi \mathrm{d}x,
\end{align*}
then
\begin{equation}
\lambda_2\int v \psi \mathrm{d}x\geq k_2\int v \psi \mathrm{d}x. \label{eqm4}
\end{equation}
In the same way, we have
\begin{align*}
\lambda_1\int u \phi \mathrm{d}x
&= \int u \Delta^2\psi \mathrm{d}x=\int \Delta^2u \psi \mathrm{d}x \\
&= \int g(v)\psi \mathrm{d}x\geq k_2 \int v \psi \mathrm{d}x\geq
\lambda_2 \int v \psi \mathrm{d}x.
\end{align*}
Or
\begin{align*}
\lambda_2\int v \psi \mathrm{d}x
&= \int v \Delta^2\phi \mathrm{d}x=\int \Delta^2v \phi \mathrm{d}x \\
&= \int f(u)\phi dx\geq k_1\int u \phi \mathrm{d}x,
\end{align*}
then
\begin{equation}
\lambda_1\int u \phi \mathrm{d}x\geq k_1\int u \phi
\mathrm{d}x. \label{eqm5}
\end{equation}
Since the integrals $\int u \phi dx$ and $\int v \psi dx$ are nonzero
and $k_1> \lambda_1$, $k_2> \lambda_2$,
the inequalities \eqref{eqm4} and \eqref{eqm5} give a contradiction.
Therefore, there exists a constant $\mu_{0}>0$ such that
\begin{equation}
w(t)\neq F(w, \mu)(t),\quad \text{for all $w \in C$ and $\mu\geq \mu_{0}$}.
\label{eqm6}
\end{equation}
This prove the third condition of (b). Now,  to prove the
second condition of (b), we choose the family of nonlinearities
$\left(f(y+\mu), g(z+\mu)\right)$ for $\mu \in [0, \mu_{0}]$, using
the a priori estimates (H1) which does not depend on $\mu$. Thus,
choosing $R> r$ large enough, we have
\begin{equation}
w(r)\neq F(w, \mu)(r),\quad  \text{for all $\mu \in [0,
\mu_{0}]$,\quad $w \in C$, \quad $\|w\|=R$}. \label{eqm7}
\end{equation}
The relations \eqref{eqm6} and \eqref{eqm7} prove the second
condition of (b).

Now, since all conditions of Theorem \ref{thfixed} are satisfied,
we apply Theorem \ref{thfixed} and we conclude the existence of a
nontrivial positive solution of problem \eqref{eqp3} and so the
existence of positive solution of problem \eqref{eq1}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{th2}]
We give this proof in four steps.
\smallskip

\noindent\textbf{Step 1.} We claim that there exist positive constants
$C_1,\dots, C_4$ such that
\begin{gather}
 \int _{B}f(u)\phi \mathrm{d}x \leq C_1,\quad
\int_{B}g(v)\psi \mathrm{d}x\leq C_2, \label{eqst11} \\
 \int _{B} u\phi \mathrm{d}x \leq C_{3},\quad \int
_{B}v \psi \mathrm{d}x\leq C_4. \label{eqst12}
\end{gather}
Indeed, using the first and second equations of \eqref{eq1}, we write
\[
 \int_{B}f(u)\phi \mathrm{d}x
= \int_{B} \Delta^2 v \phi \mathrm{d}x
= \int_{B}v\Delta^2\phi\mathrm{d}x=\lambda_2\int_{B}v\psi \mathrm{d}x.
\]
From condition (i) of Theorem \ref{th1}, there exist $k_2> \lambda_2$ and
$A> 0$ such that $g(v)\geq k_2 v-A$. Thus, for a positive constant $C$, we have
\begin{equation}
 \int_{B}f(u) \phi \mathrm{d}x =
\lambda_2\int_{B}v\psi \mathrm{d}x\leq C+
\frac{\lambda_2}{k_2}\int_{B}g(v) \psi \mathrm{d}x .
\label{eqst13}
\end{equation}
In the same way, we have
\[
\int_{B}g(v)\psi \mathrm{d}x
=\int_{B} \Delta^2 u \psi \mathrm{d}x
=\int_{B}u\Delta^2\psi \mathrm{d}x=\lambda_1\int_{B}v\phi \mathrm{d}x.
\]
Again, from condition (i) of Theorem \ref{th1}, there exist $k_1> \lambda_1$
and $D> 0$ such that $f(u)\geq k_1 u-D$. Thus, for a positive constant $\bar{C}$,
we have
\begin{equation}
 \int_{B}g(v)\psi \mathrm{d}x
=\lambda_1\int_{B}u\phi \mathrm{d}x\leq \bar{C}+
\frac{\lambda_1}{k_1}\int_{B}f(u) \phi \mathrm{d}x .
\label{eqst14}
\end{equation}
Combining \eqref{eqst13} and \eqref{eqst14}, for  positive constants $M_1$
and $M_2$, we obtain
\begin{gather*}
 \int_{B}f(u) \phi \mathrm{d}x\leq M_1+
\frac{\lambda_1\lambda_2}{k_1k_2}\int_{B}f(u) \phi \mathrm{d}x, \\
 \int_{B}g(v) \psi \mathrm{d}x\leq M_2+
\frac{\lambda_1\lambda_2}{k_1k_2}\int_{B} g(v) \psi
\mathrm{d}x.
\end{gather*}
Since $\frac{\lambda_1\lambda_2}{k_1k_2}<1$, we deduce \eqref{eqst11}.
Using, again, condition (i) of Theorem \ref{th1} and \eqref{eqst11} we
easily deduce \eqref{eqst12}.
\smallskip

\noindent \textbf{Step 2.} We claim that there exist positive constants
$C_1$,\dots, $C_4$ such that
\begin{gather}
u(r)\leq C_1,\quad v(r)\leq C_2, \quad \text{for $\frac{2}{3}\leq r \leq  1$,}
\label{eqst21} \\
u''(1)\leq C_{3},\quad v''(1)\leq C_4.
\label{eqst22}
\end{gather}
Indeed, we see that
\begin{gather*}
 u(r)=\int_{0}^{1}G(r,s)g(v(s))\mathrm{d}s, \\
 v(r)=\int_{0}^{1}G(r,s)f(u(s))\mathrm{d}s.
\end{gather*}
Using the fact that $r\to G(r,s)$ is decreasing (see
\eqref{eqg1} and \eqref{eqg2}), we deduce that $u(r)$ and
$v(r)$ are decreasing in $r$ and, for arbitrary
$\frac{2}{3}\leq r \leq 1$,
\begin{align*}
u(r)
&\leq u(\frac{2}{3})
\leq 3\int_{1/3}^{2/3}u(s)\mathrm{d}s \\
&\leq C\int_{0}^{1}s^{N-1}(1-s)^2u(s)\mathrm{d}s
\leq C+\int_{0}^{1}s^{N-1}(1-s)^2u(s)\mathrm{d}s.
\end{align*}
Looking at \eqref{eqvalue} and Lemma \ref{lm2}, we write
\begin{equation*}
u(r)\leq C\Big(1+\int_{0}^{1}s^{N-1}(1-s)^2u(s)\mathrm{d}s\Big)
\leq  C\Big(1+\int_{B}\phi u \mathrm{d}x\Big).
\end{equation*}
We conclude by \eqref{eqst12} that $ u(r)\leq C_1$, for
$\frac{2}{3}\leq r \leq 1$ and by the same way that $ v(r)\leq C_2$
for $\frac{2}{3}\leq r \leq 1$.

Now, we prove \eqref{eqst22}. We have
\[
u(r)=\int_{0}^{1}G(r,s)g(v(s))\mathrm{d}s,\quad
v(r)=\int_{0}^{1}G(r,s)f(u(s))\mathrm{d}s.
\]
We differentiate the two previous relations two times with respect
to $r$, we obtain
\begin{equation*}
u''(r)=\int_{0}^{1}\frac{\partial^2 G(r,s)}{\partial
r^2}g(v(s))\mathrm{d}s,\quad
v''(r)=\int_{0}^{1}\frac{\partial^2 G(r,s)}{\partial
r^2}f(u(s))\mathrm{d}s.
\end{equation*}
Since the integrals converge, we take the limit when $r$ approaches
$1$, we write
\begin{equation*}
u''(1)=\int_{0}^{1}\frac{\partial^2 G(r,s)}{\partial
r^2}{\big{|}_{r=1}}g(v(s))\mathrm{d}s\quad
v''(1)=\int_{0}^{1}\frac{\partial^2 G(r,s)}{\partial
r^2}{\big{|}_{r=1}}f(u(s))\mathrm{d}s.
\end{equation*}
Using \eqref{eqg3}, we obtain
\begin{equation*}
u''(1)=\frac{1}{2}\int_{0}^{1}s^{N-1}(1-s^2)g(v(s))\mathrm{d}s,\quad
v''(1)=\frac{1}{2}\int_{0}^{1}s^{N-1}(1-s^2)f(u(s))\mathrm{d}s.
\end{equation*}
From \eqref{eqvalue} and Lemma \ref{lm2}, we see, for some
positive constant $C$, that
\begin{equation*}
u''(1)\leq C \int_{B}\psi g(v) \mathrm{d}x,\quad
v''(1)\leq C \int_{B}\phi f(u) \mathrm{d}x.
\end{equation*}
Finally, we obtain \eqref{eqst22} by \eqref{eqst11}.
\smallskip

\noindent\textbf{Step 3.}
Under conditions (i) and (ii) of Theorem \ref{th1}, we claim that, for a
small number $0<l<1$, there exist positive constants
$C_1, \dots, C_4$ such that
\begin{gather}
\int_{0}^{l}s^{N-1} f(u(s))  \mathrm{d}s \leq C_1, \quad
\int_{0}^{l}s^{N-1}  g(v(s)) \mathrm{d}s \leq C_2, \label{eqst31} \\
\int_{B}u f(u) \mathrm{d}x \leq C_{3}, \quad
\int_{B}v g(v) \mathrm{d}x \leq C_4. \label{eqst32}
\end{gather}
Indeed, for the proof of \eqref{eqst31}, looking at proof of Step 1,
 namely at \eqref{eqst11} and \eqref{eqst12}, and using Lemma \ref{lm1}
 and Lemma \ref{lm2} we obtain, for small $0<l <1$,
\begin{align*}
\int_{0}^{l} s^{N-1} f(u(s)) \mathrm{d}s
&\leq  \int_{0}^{l}s^{N-1}\frac{(1-s)^2}{(1-l)^2}f(u(s))\mathrm{d}s\\
&\leq  \frac{1}{(1-l)^2} \int_{0}^{l}s^{N-1}(1-s)^2 f(u(s))\mathrm{d}s\\
&\leq   C \int_{0}^{1}s^{N-1}\phi(s) f(u(s))\mathrm{d}s \\
&=C \int_{B} f(u) \phi \mathrm{d}x \leq M_1,
\end{align*}
and
\begin{align*}
\int_{0}^{l} s^{N-1} g(v(s)) \mathrm{d}s
&\leq  \int_{0}^{l}s^{N-1}\frac{(1-s)^2}{(1-l)^2}g(v(s))\mathrm{d}s\\
&\leq  \frac{1}{(1-l)^2} \int_{0}^{l}s^{N-1} (1-s)^2 g(v(s))\mathrm{d}s\\
&\leq  \bar{C}\int_{0}^{1}s^{N-1}\psi(s) g(v(s))\mathrm{d}s\\
&=\bar{C} \int_{B} g(v) \psi \mathrm{d}x \leq M_2,
\end{align*}
where $C$, $\bar{C}$, $M_1$ and $M_2$ are some constants.
This gives \eqref{eqst31}.

For the proof of \eqref{eqst32}, we rewrite the identity \eqref{idserrin}
of Lemma \ref{pohozaev}, considering the fact that $\alpha+\beta=N-4$, as
\begin{equation*}
\int_{B}N F(u)-\alpha u f(u)\mathrm{d}x+\int_{B}N
G(v)-\beta v g(v) \mathrm{d}x
=\int_{\partial B}(\Delta u, \Delta v)  (x. \nu)\mathrm{d}\sigma_{x}.
\end{equation*}
Using condition (ii) of Theorem \ref{th1} for the left hand side of
the last equality and after easy computation on the right hand side, we obtain
\begin{equation*}
 \theta_1\int_{B}u f(u)dx+\theta_2\int_{B}v g(v) dx\leq C u''(1)v''(1),
\end{equation*}
where $C$ is a generic constant and $\theta_1$, $\theta_2$ are the constants
given by hypothesis of Theorem \ref{th1}. Therefore
\begin{equation*}
\theta_1\int_{B}u \,f(u) dx+\theta_2\int_{B}v\, g(v) dx\leq C .
\end{equation*}
Since the two both left hand sides are positive we obtain directly \eqref{eqst32}.
\smallskip

\noindent\textbf{Step 4.}
Under  conditions (i) and (ii) of
Theorem \ref{th1}, we claim that there exist positive constants
$C_1$ and $C_2$ such that, for any solution $(u, v)$ of
problem \eqref{eq1},
\begin{equation}
\|u\|_{\infty}\leq C_1, \quad \|v\|_{\infty}\leq C_2.\label{eqst41}
\end{equation}
Indeed, for $u$, we have
\begin{align*}
\|u\|_{\infty}
&\leq  u(0)\leq \int_{0}^{1}G(0, s)\,g(v(s)) \mathrm{d}s\\
&\leq  C \int_{0}^{1} s^{3}(1-s)^2\,g(v) \mathrm{d}s
 \leq C \int_{0}^{1} s^{3}\,g(v) \mathrm{d}s\\
&\leq  C\int_{0}^{t}s^{3}\, g(v(s)) \mathrm{d}s
 +C \int_{t}^{1} s^{3} \, g(v)\mathrm{d}s,
\end{align*}
where $t \in (0, 1)$ is arbitrary and $C$ denotes a positive constant whose
value may vary from line to line.

Let $g(m):= \max_{s\in [0,m]} g(s) $ for $m \in (0,\infty)$,
applying H\"older's inequality, we obtain
\begin{align*}
\|u\|_{\infty} 
&\leq  C t^{4} g(\|v\|_{\infty})
 +C\Big(\int_{t}^{1}s^{\gamma_1(q^{*}+1)}\mathrm{d}s \Big)^{\frac{1}{q^{*}+1}}
 \Big(\int_{t}^{1}s^{N-1} (g(v(s)))^{\frac{q^{*}+1}{q^{*}}}\mathrm{d}s
 \Big)^{\frac{q^{*}}{q^{*}+1}}\\
&\leq  C t^{4}\, g(\|v\|_{\infty})
 +C\Big(\int_{t}^{1}s^{\gamma_1(q^{*}+1)}\mathrm{d}s \Big)^{\frac{1}{q^{*}+1}} \\
&\quad\times \Big(\int_{t}^{1}s^{N-1} g(v(s)) (g(v(s)))^{\frac{1}{q^{*}}}\mathrm{d}s
 \Big)^{\frac{q^{*}}{q^{*}+1}},
\end{align*}
where $\gamma_1=3-(N-1)\frac{q^{*}}{q^{*}+1}$. Or from Remark \ref{rm1},
there exists a positive constant $M$ such that
\begin{equation}
\begin{gathered}
 g(s)< M (1+s)^{q^{*}},\quad \text{for all $s\geq 0$,}\\
 f(s)< M (1+s)^{p^{*}},\quad \text{for all $s\geq 0$}.
\end{gathered}\label{ft}
\end{equation}
Then
\begin{align*}
\|u\|_{\infty} 
&\leq   C t^{4} g(\|v\|_{\infty})+M^{\frac{1}{q^{*}}}
 C\Big(\int_{t}^{1}s^{\gamma_1(q^{*}+1)}\mathrm{d}s \Big)^{\frac{1}{q^{*}+1}} \\
&\quad\times  \Big(\int_{t}^{1}s^{N-1} g(v(s))\,(1+v(s))\mathrm{d}s\Big)^{\frac{q^{*}}{q^{*}+1}}\\
&\leq   C t^{4} g(\|v\|_{\infty})+M^{\frac{1}{q^{*}}}
 C\Big(\int_{t}^{1}s^{\gamma_1(q^{*}+1)}\mathrm{d}s \Big)^{\frac{1}{q^{*}+1}}\\
&\quad\times  \Big(\int_{B} g(v) \mathrm{d}x+\int_{B} g(v)\, v(x)\mathrm{d}x
 \Big)^{\frac{q^{*}}{q^{*}+1}}.
\end{align*}
Using \eqref{eqst31} and \eqref{eqst32}, we obtain
\begin{equation*}
\|u\|_{\infty}\leq C  t^{4}\, g(\|v\|_{\infty})
+C\Big(\int_{t}^{1}s^{\gamma_1(q^{*}+1)}\mathrm{d}s \Big)^{\frac{1}{q^{*}+1}}.
\end{equation*}
In a similar way,  for $v$, we have
\begin{equation*}
\|v\|_{\infty}\leq C t^{4} f(\|u\|_{\infty})
+C\Big(\int_{t}^{1}s^{\gamma_2(p^{*}+1)}\mathrm{d}s \Big)^{\frac{1}{p^{*}+1}},
\end{equation*}
where $\gamma_2=3-(N-1)\frac{p^{*}}{p^{*}+1}$.

In all the next inequalities $C$ will always represent a positive constant,
 not necessarily the same in each occurrence. After some computations, we obtain
\begin{gather}
\|u\|_{\infty}\leq C t^{4} g(\|v\|_{\infty})+Ct^{\frac{4+(4-N)q^{*}}{q^{*}+1}},
\label{eqst42} \\
\|v\|_{\infty}\leq C  t^{4} f(\|u\|_{\infty})+C t^\frac{4+(4-N)p^{*}}{p^{*}+1}.
\label{eqst43}
\end{gather}
Note that if $g$ and $f$ are bounded then \eqref{eqst41} comes immediately.
However, if $g$ is not bounded then, by Remark \ref{rm1}, there exists a
 positive $K$, see \eqref{ft}, such that $  g(r)\leq K r^{q^{*}}$, for $r\geq 1$.
Therefore, we can write \eqref{eqst42} as
\begin{equation}
\|u\|_{\infty}\leq C  t^{4} (\|v\|_{\infty})^{q^{*}}+C t^{\frac{4+(4-N)q^{*}}{q^{*}+1}}.
\label{eqst44}
\end{equation}
Now, inserting \eqref{eqst43} into \eqref{eqst44}, and using the inequality
$(a+b)^{m}\leq C_{m} (a^{m}+\,b^{m})$ for $a,b,m \geq 0$ where $C_{m}$ is a
 positive constant depending of $m$, we obtain
\begin{equation}
\|u\|_{\infty}\leq C  t^{4(q^{*}+1)}\,(f(\|u\|_{\infty}))^{q^{*}}
+C  t^{\frac{[4+(4-N)p^{*}]q^{*}}{p^{*}+1}+4}
+C t^{\frac{4+(4-N)q^{*}}{q^{*}+1}}.
\label{eqst45}
\end{equation}
Easy computations show that
\begin{equation*}
\frac{[4+(4-N)p^{*}]q^{*}}{p^{*}+1}+4
=\frac{4+(4-N)q^{*}}{q^{*}+1}=-\frac{N}{p^{*}+1}
=4-\frac{Nq^{*}}{q^{*}+1}.
\end{equation*}
Therefore,
\begin{equation}
\|u\|_{\infty}\leq C  t^{4(q^{*}+1)}(f(\|u\|_{\infty}))^{q^{*}}
+C  t^{4-\frac{Nq^{*}}{q^{*}+1}}.
\label{eqst46}
\end{equation}
Let $r= 4 q^{*}\big[1+\frac{1}{(p^{*}+1)(q^{*}+1)}\big]$.
Since $t\in [0,\,1]$, we write \eqref{eqst46} as
\begin{equation*}
\|u\|_{\infty}\leq C  t^{r}\,(f(\|u\|_{\infty}))^{q^{*}}
+C  t^{4-\frac{Nq^{*}}{q^{*}+1}}.
\end{equation*}
Let $h(t)= t^{r}(f(\|u\|_{\infty}))^{q^{*}}+ t^{4-\frac{Nq^{*}}{q^{*}+1}}$.
The function $h$ attains its infimum at
\[
t_{0}= C (f(\|u\|_{\infty}))^{-\frac{q^{*}(q^{*}+1)}{(r-4)(q^{*}+1)+Nq^{*}}}
\]
 and we have
\begin{equation*}
\|u\|_{\infty}\leq C \,(f(\|u\|_{\infty}))^{\frac{-q^{*}(q^{*}+1)r}{(r-4)(q^{*}+1)
+N q^{*}}+q^{*}}+C (f(\|u\|_{\infty}))^{\frac{-q^{*}(q^{*}+1)}{(r-4)(q^{*}+1)
+N q^{*}}(4-\frac{N q^{*}}{q^{*}+1})}.
\end{equation*}
Some computation show that
$$
\frac{-q^{*}(q^{*}+1)r}{(r-4)(q^{*}+1)+N q^{*}}+q^{*}
=  \frac{- q^{*}(q^{*}+1)}{(r-4)(q^{*}+1)+N q^{*}}(4-\frac{N q^{*}}{q^{*}+1})
=\frac{1}{p^{*}}.
$$
Therefore
\begin{equation*}
\|u\|_{\infty}\leq C  (f(\|u\|_{\infty}))^{1/p^{*}}.
\end{equation*}
Or Remark \ref{rm1} implies that $f(x)=o(x^{p^{*}})$ for $x\to \infty$
then the last inequality becomes
\begin{equation*}
\|u\|_{\infty}\leq C (1+o(\|u\|_{\infty})),
\end{equation*}
which proves that $\|u\|_{\infty}$ is bounded and then implies,
by  \eqref{eqst43}, that $\|v\|_{\infty}$ is bounded.
This completes Step 4.
\end{proof}


\begin{proof}[Proof of Theorem \ref{th3}]
In the identity \eqref{idserrin}, we take $\alpha+\beta=N-4$.
Since $u=0=\frac{\partial u}{\partial \nu}$ and
$v=0=\frac{\partial v}{\partial \nu}$, we have
$(\Delta u, \Delta v)=\frac{\partial^2u}{\partial \nu^2}
\frac{\partial^2v}{\partial \nu^2}$.
If $(u, v)$ is a non trivial solution of \eqref{eq1}, since $B$
is star-shaped domain about $0$, then $x.\nu \geq 0$ on $\partial B$
and the identity \eqref{idserrin} gives a contradiction when
\begin{equation*}
N F(s)-\alpha s f(s)\leq 0\quad \text{and} \quad N G(s)-\beta s g(s)\leq 0.
\end{equation*}
The proof is complete.
\end{proof}

\subsection*{Acknowledgments}
This work was supported by the Deanship of the Scientific Research of Jazan
University, Saudi Arabia (Project 837 - Sabic 2 - 36).

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\end{document}