\documentclass[twoside]{article} \usepackage{amssymb} % font used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil Four-parameter bifurcation \hfil EJDE--2001/06} {EJDE--2001/06\hfil J. Fleckinger, R, Pardo, \& F. de Th\'elin \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2001}(2001), No. 06, pp. 1--15. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Four-parameter bifurcation for a \\ p-Laplacian system % \thanks{ {\em Mathematics Subject Classifications:} 35J45, 35J55, 35J60, 35J65, 35J30, 35P30 \hfil\break\indent {\em Key words:} p-Laplacian, bifurcation. \hfil\break\indent \copyright 2001 Southwest Texas State University. \hfil\break\indent Submitted June 29, 2000. Published January 9, 2001. \hfil\break\indent (R.P.) supported by grant PB96-0621 from the Spanish DGICYT } } \date{} % \author{ Jacqueline Fleckinger, Rosa Pardo, \& Fran\c{c}ois de Th\'elin } \maketitle \begin{abstract} We study a four-parameter bifurcation phenomenum arising in a system involving $p$-Laplacians: $$\displaylines{ -\Delta_p u = a \phi_p(u)+ b \phi_p(v) + f(a , \phi_p (u), \phi_p (v)) ,\cr -\Delta_p v = c \phi_p(u) + d \phi{p}(v)) + g(d , \phi_p (u), \phi_p (v)), }$$ with $u=v=0$ on the boundary of a bounded and sufficiently smooth domain in $\mathbb{R}^N$; here $\Delta_{p}u = {\rm div} (| \nabla u|^{p-2} \nabla u)$, with $p>1$ and $p \neq 2$, is the $p$-Laplacian operator, and $\phi_{p} (s) =|s|^{p-2} s$ with $p>1$. We assume that $a, b, c, d$ are real parameters. Thwn we use a bifurcation method to exhibit some nontrivial solutions. The associated eigenvalue problem, with $f=g \equiv 0$, is also studied here. \end{abstract} % \newtheorem{theorem}{Theorem} [section] \newtheorem{proposicion}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{lemma}[theorem]{Lemma} \renewcommand{\theequation}{\thesection.\arabic{equation}} \catcode`@=11 \@addtoreset{equation}{section} \catcode`@=12 \section{Introduction and Hypotheses} We study some four-parameter bifurcation phenomena arising in the system \begin{eqnarray} \label{p-System} &- \Delta_p u = a \phi_p(u)+ b \phi_p(v) + f(a , \phi_p (u), \phi_p (v)),&\nonumber \\ & - \Delta_p v = c \phi_p(u) + d \phi{p}(v) + g(d , \phi_p (u), \phi_p (v)), \quad\mbox{in }\Omega & \\ &u = v = 0\,, \quad\mbox{on }\partial \Omega\,.& \nonumber \end{eqnarray} where $ \Delta_pu = \mathop{\rm div} (| \nabla u|^{p-2} \nabla u)$ for $p>1, \ p \neq 2$, is the $p$-Laplacian operator, $\phi_p :\mathbb{R}\to \mathbb{R}$ is given by $\phi_p (s) = |s|^{p-2} s, \ p>1$, $\Omega \subset \mathbb{R}^{N}$ is a sufficiently smooth bounded domain, and $a, b ,c, d $ are real parameters. The operator $ {-\Delta}_p$ occurs in problems arising in pure mathematics, such as the theory of quasiregular and quasiconformal mappings (see \cite{To} and the references therein), and in a variety of applications, such as non-Newtonian fluids, reaction-diffusion problems, flow through porous media, nonlinear elasticity, glaciology, petroleum extraction, astronomy, etc (see \cite{At-EA,At-Ch,Di,Ar-Di}). We also emphasize that systems such as (\ref{p-System}) are not easy generalizations of equations because the solutions cannot be obtained by variational methods. Here we use a bifurcation method to exhibit some nontrivial solutions. Another approach for non variational systems can be found in \cite{C-F-M-T}. Moreover the problem considered here where $p \neq 2$ is not a straightforward extension of the case $p=2$ due to the fact that the translations of the p-Laplacian are not always invertible neither commutative. In this paper we obtain bifurcation results for (\ref{p-System}). The linear case ($p=2$) is studied in \cite{F-P}. The case where $g\equiv 0$ is considered in \cite{F-M-T}. We assume through this article that the functions $f$ and $g$ satisfy the following Hypothesis: A continuous function $f:\mathbb{R}^3 \to \mathbb{R}$ satisfies Hypothesis {\bf (H)} if there exists $\rho$ such that $1 \leq \rho < {\frac{N+p'}{N-min(p,p')}}$ for $\min (p,p')1$, $q'$ is defined by: $$\frac{1}{q} + \frac{1}{q'}=1.$$ \paragraph{Definitions:} By a solution of the system (\ref{p-System}) we mean a pair $(A,(u,v))\in \mathbb{R}^4 \times (W_0^{1,p}(\Omega) )^2$, with $A:= \left(\begin{array}{cc} a & b \\ c & d\end{array}\right)$, satisfying (\ref{p-System}) in the weak sense, i.e., for all $w, z \in W_0^{1,p}(\Omega)$, \begin{eqnarray} \label{weak.p-System} \int _{\Omega} | \nabla u|^{p-2} \nabla u . \nabla w & = & \int _{\Omega} a |u|^{p-2}u w + b |v|^{p-2}v w + f(a , \phi_p (u), \phi_p (v)) w \nonumber \\ \int _{\Omega}| \nabla v|^{p-2} \nabla v .\nabla z & = & \int _{\Omega} c |u|^{p-2}uz + d |v|^{p-2}v z + g(d , \phi_p (u), \phi_p (v)) z \end{eqnarray} The set of solutions will be denoted by ${\cal S}$. Obviously $(A,(0,0))$ is a solution of (\ref{p-System}) for every $ (a ,b,c, d) \in \mathbb{R}^4$. The set of these pairs will be called the trivial solution set, and will be denoted by ${\cal S}_0$. We say that $(A_0,(0,0)) \in {\cal S}_0$ is a {\em bifurcation point} of (\ref{p-System}) with respect to the trivial solution set iff every neighborhood of $(A_0,(0,0))$ contains solutions of (\ref{p-System}) belonging to ${\cal S} \setminus {\cal S}_0$. We will show that whenever (H) is satisfied, any matrix $A_0$ with a negative eigenvalue, the other being the principal eigenvalue of the $p$-Laplacian, is such that $(A_0,(0,0)) \in {\cal S}_0$ is a bifurcation point to positive solutions for (\ref{p-System}). To establish our results, we combine and adapt methods of \cite{F-P} and \cite{F-M-T}. Our paper is organized as follows: In Section 2, we recalls some results concerning the $p$-Laplacian. We recall in particular several lemmas established in \cite{F-M-T} concerning spaces that we will use. In Section 3, we show that if $(A_0,(0,0)) \in {\cal S}_0$ is a bifurcation point, then the homogeneous system: $-\Delta_pU=A_0U$ has a non trivial solution. In Section 4 we obtain conditions on $A_0$ for this to happen. In Section 5 we compute the Leray-Schauder degree for the eigenvalue problem and in Section 6 we state and establish our result. \section{Notation and preliminaries} In this section, we recall briefly some notation and results concerning the p-Laplacian. \noindent {\bf The $p$-Laplacian}, $-\Delta_p$, defined on $W_0^{1,p}(\Omega)$ has a first eigenvalue $\lambda_1(p):=\lambda_1$ which is simple and isolated \cite{An}; it is associated to a simple eigenfunction $\varphi$ (normalized as $\|\varphi\|_{\infty}=1$) which is positive. Moreover, $\lambda_1$ is characterized by \begin{equation} \label{vp1} \lambda_1 = \inf_{u\in W^{1,p}_0; \int_{\Omega} |u|^p = 1} \int_{\Omega}\mid \nabla u\mid^p\,. \end{equation} The following results are known for the equation \begin{eqnarray} \label{p-Eq} &-\Delta_pu \, = \, k|u|^{p-2}u +f \quad\mbox{in } \Omega &\\ &u=0 \quad\mbox{on } \partial \Omega\,\label{cl}.& \end{eqnarray} \begin{lemma}[{\cite{V}}] \label{PM} If $f\in L^{\infty}(\Omega)$, $f\geq 0$, $f\not \equiv 0$, Equation (\ref{p-Eq}-\ref{cl}) has at least one solution and satisfies the maximum principle (i.e. any solution $u$ is non-negative) if and only if $k<\lambda_1$. \end{lemma} \begin{lemma}[{\cite{F-G-T-T}}] For $f\in L^{\infty}$, $f\geq 0$, $f\not \equiv 0$, and for $k=\lambda_1$, Equation (\ref{p-Eq}-\ref{cl}) has no solution in $ W_0^{1,p}(\Omega)$. \end{lemma} \paragraph{The operator $T_q$.} We introduce now some notation and results used in \cite{F-M-T}. Let \begin{equation}{\cal A}(q)= \left\{ \begin{array}{ll} {\frac{Nq'}{N-min(q,q')}} & \mbox{ if } \min (q,q')0$. We have the following embeddings. \begin{lemma}[{Lemma 2.2 in \cite{F-M-T}}] If $\alpha < {\cal A}(q)$ the embedding $\phi_q(W^{1,q})$ into $L^{\alpha}$ is compact. If $\beta > {\cal B}(q)$, the embedding $L^{\beta}$ into $W^{-1,q'}$ is compact. \end{lemma} \begin{lemma}[{\cite{F-M-T}}] For $\alpha < {\cal A}(q)$, $\beta > {\cal B}(q)$, and $k<0$, the operators $$T_q:D(T_q)\subset L^{\alpha} \longrightarrow L^{\beta}\quad\mbox{and}\quad (T_q-k)^{-1}: L^{\beta} \longrightarrow L^{\alpha}$$ are well defined and $(T_q-k)^{-1}: L^{\beta} \longrightarrow L^{\alpha}$ is completely continuous. \end{lemma} \begin{lemma} {\it For $\alpha < {\cal A}(q)$, $\beta > {\cal B}(q)$, $k<\lambda_1$ and $f\in L^{\beta}, f>0$, $(T_q-k)^{-1}f$ is well defined or equivalently (\ref{p-Eq}) has a unique solution.}\end{lemma} \begin{remark} \rm Obviously, $\Delta_q (-u)= -\Delta_q u$ and $\phi_q(-s)= -\phi_q(s)$, then it follows that $T_q(-u)=-T_q(u)$ and by the previous Lemma with $k<\lambda_1$ and $f\in L^{\beta}, f<0$, $(T_q-k)^{-1}f$ is also well defined. When $f$ changes sign several solutions may appear \cite{P-E-M,FHTT2}. \end{remark} \noindent We also introduce $$ a(q)= \left\{ \begin{array}{ll} \frac{Nq}{N-q} & \mbox{ if } q1$, we have $$ (a(q))'0$, there exists a constant $C$ such that $$ |F(\lambda , r,s)| \leq \delta + C|(r,s)|^{\rho}, \quad \forall (r,s) \in \mathbb{R}\times \mathbb{R}\,; $$ hence the first assertion holds. Now, by H\"older‘ s inequality, \begin{eqnarray} \label{inequal} \lefteqn{ \int _{\Omega } \Big| \frac{F(\lambda ,w_n ,z_n )}{ \| (w_n,z_n) \|_{L^{\alpha }\times L^{\alpha } }} \Big|^{\beta } }\nonumber\\ &\leq& \Big( \int_{\Omega } \big| \frac{F(\lambda ,w_n ,z_n )}{ | (w_n,z_n) |} \big|^{\frac{\alpha }{\rho -1} }\Big)^{1/\rho'} \Big( \int _{\Omega } \big| \frac{|(w_n ,z_n )|}{ \| (w_n,z_n) \|_{L^{\alpha } \times L^{\alpha } }} \big|^{\alpha } \Big)^{1/\rho} \nonumber\\ &\leq& C_1 \Big( \int _{\Omega } \big| \frac{F(\lambda ,w_n ,z_n )}{ | (w_n,z_n) |} \big|^{\alpha/(\rho -1)}\Big)^{1/\rho'}. \end{eqnarray} From (H) we deduce $$ \left| \frac{F(\lambda ,w_n ,z_n )}{ | (w_n,z_n) |} \right|^{\frac{\alpha }{\rho -1} } \leq \delta ^{\frac{\alpha }{\rho -1}}+C_2 |(w_n ,z_n )|^{\alpha }. $$ Since $ \displaystyle \lim_{n\to \infty } \| (w_n,z_n) \|_{L^{\alpha }\times L^{\alpha }}=0$, for every $\delta >0$, $$ \limsup_{n\to \infty } \int _{\Omega } \left| \frac{F(\lambda ,w_n ,z_n ) }{ | (w_n,z_n) |} \right|^{\frac{\alpha }{\rho -1} } \leq \delta ^{\frac{\alpha }{\rho -1}} |\Omega |\,. $$ Taking into account (\ref{inequal}) the results follows. \hfill$\diamondsuit$ \section{Preliminary results} In this section we show that if $(A_0,(0,0))$ is a bifurcation point, then the eigenvalue problem \begin{eqnarray}\label{p-EP} &- \Delta_p u = a_0 \phi_p(u) + b_0 \phi_p(v), &\nonumber \\ &- \Delta_p v = c_0 \phi_p(u) + d_0 \phi_p(v), \quad \mbox{in } \Omega \\ &u = v = 0, \quad\mbox{on } \partial \Omega \,. \nonumber \end{eqnarray} has a non-trivial solution. This is well-known in the case $p=2$, (cf. \cite{C-R}), but due to the nonlinearity of $T_p$, the proof is much more delicate. \begin{theorem} \label{bif.p-EP} Let $f,g$ satisfy $(H1)$, and $(A_0 , (0,0))$ be a bifurcation point of (\ref{p-System}) in $\mathbb{R}^4 \times (W_0^{1,p}(\Omega) )^2$ ; then the eigenvalue problem (\ref{p-EP}) has a non-trivial solution . \end{theorem} \paragraph{Proof.} If $(A_0, (0,0))$ is a bifurcation point, then there exists a sequence \\ $\{(A_n, (u_{n},v_{n})) \}$ of nontrivial solutions of (\ref{p-System}), with $A_n=(a_{n} , b_{n},c_{n}, d_n)\in \mathbb{R}^4$ and $ (u_{n},v_{n})\in (W_0^{1,p}(\Omega))^2$, such that $$ A_{n} \to A_0 \quad\mbox{in } \mathbb{R}^4 \quad\mbox{and}\quad (u_{n},v_{n}) \to (0,0) \quad\mbox{in } (W_0^{1,p}(\Omega))^2. $$ Define $w_{n} = \phi_p (u_{n}), \ z_{n}= \phi_p(v_{n})$. Due to Lemma 2.2 (cf. \cite{F-M-T} ; Lemma 2.2), $w_{n}, \ z_{n} \in L^{\alpha}$ whenever $\alpha < {\cal A}(p)$. Moreover, $(A_{n} , (w_{n},z_{n}))$ is a nontrivial solution of the system \begin{eqnarray} \label{p-TSystem} &T_p w_{n} = a_{n} w_{n} + b_{n} z_{n} + f(a_{n} , w_{n}, z_{n} ) ,& \\ &T_p z_{n} = c_{n} w_{n} + d_{n} z_{n} + g(d_{n} , w_{n} , z_{n} ) \quad\mbox{in }\Omega . & \nonumber \end{eqnarray} Let $s_{n} = \max \{ \| w_{n}\|_{L^{\alpha}}, \| z_{n}\|_{L^{\alpha}} \}>0$. By Lemma 2.2 above it is obvious that $s_{n} \to 0$ as $n \to \infty$. We define $$ W_{n} = \frac{w_{n}}{s_{n}} , \quad Z_{n}= \frac{z_{n}}{s_{n}}, \quad n \in \mathbb{N} $$ Dividing each equation of System (\ref{p-TSystem}) by $s_{n}$ we can write \begin{eqnarray} &W_{n} = T_p^{-1}\left( a_{n} W_{n} + b_{n} Z_{n} + \frac{1}{s_{n}} f(a_{n} , w_{n}, z_{n} ) \right) ,&\nonumber \\ &Z_{n} = T_p^{-1} \left(c_{n} W_{n} + d_{n} Z_{n} + \frac{1}{s_{n}} g(d_{n} , w_{n} , z_{n} ) \right), \nonumber \quad\mbox{in } \Omega\,.& \end{eqnarray} From Lemma \ref{imbedding}, $ f(\mathbb{R}\times L^{\alpha} \times L^{\alpha} ) \subset L^{\beta}$ for $\beta = \frac{\alpha}{\rho}$ and $$ \limsup_{n \to \infty} \left\| \frac{f(a_{n} , w_{n}, z_{n} )}{s_n} \right\|_{L^{\beta}} = 0. $$ Of course an analogous result holds for $g$. Therefore, $$ a_{n} W_{n} + b_{n} Z_{n} + \frac{1}{s_{n}}f(a_{n} , w_{n}, z_{n} ) \quad\mbox{and}\quad c_{n} W_{n} + d_{n} Z_{n} + \frac{1}{s_{n}} g(d_{n} , w_{n} , z_{n}) $$ are bounded sequences in $L^{\beta}$ with $\beta <\alpha $. It follows from the compactness $T_p^{-1} : L^{\beta} \to L^{\alpha}$ that there exists two convergent subsequences $$\displaylines{ T_p^{-1} \Big( a_{n} W_{n} + b_{n} Z_{n} + \frac{1}{s_{n}} f(a_{n} , w_{n}, z_{n} ) \Big) \to W\,,\cr T_p^{-1} \Big( c_{n} W_{n} + d_{n} Z_{n} + \frac{1}{s_{n}} g(d_{n} , w_{n} , z_{n}) \Big) \to Z }$$ in $L^{\alpha}$ and $(W,Z) \neq (0,0)$. Moreover $(W_{n},Z_{n}) \to (W,Z)$ in $L^{\alpha}$ and \begin{eqnarray} &T_p W = a_0 W + b_0 Z,&\nonumber \\ &T_p Z = c_0 W + d_0 Z, \quad\mbox{in }\Omega ,& \nonumber \end{eqnarray} or equivalently, $(W,Z)$ is a nontrivial solution of the eigenvalue problem (\ref{p-EP}). \section{An Eigenvalue problem} In this section we consider the eigenvalue problem (\ref{p-EP}) with $(a_0 , b_0,c_0, d_0)=(a ,b,c, d)$. We establish necessary and sufficient conditions so that System $(3.1)$ has a nontrivial positive solution. \paragraph{Definition.} We say that $A = \left( \begin{array}{c c} a & b \\ c & d \end{array} \right)$ satisfies the {\it solvability condition}, and we write $A\in {\cal S}(T_p)$, if there exists a nontrivial solution of \begin{equation} \label{p-TP} T_p \left( \begin{array}{c} w \\ z \end{array} \right) = A \left( \begin{array}{c} w \\ z \end{array} \right)\,, \end{equation} where $ \ w:= \phi_p (u)$, $z=\phi_p(v)$, $w, z \in D(T_p)$ with $$ D(T_p) := \{ z \in L^{\alpha (p) }(\Omega): \phi_{p'}(z) \in W_0^{1,p}(\Omega), \; -\Delta_p(\phi_{p'}(z)) \in L^{\beta (p)} (\Omega)\}, $$ and $\alpha (p), \beta (p)$ satisfy \begin{equation} ({\cal B}(p))'< \beta (p) \leq \alpha (p) < {\cal A}(p) . \end{equation} We remark that Problem $(3.1)$ is equivalent to the operator equation (\ref{p-TP}). \paragraph{Definition.} Let $\sigma(A)$ denote the spectrum of the Matrix $A$. Let ${\cal M}^-$ be the set of matrices that have a negative eigenvalue. \begin{remark} \rm Since $A$ has real coefficients the eigenvalues are complex conjugate; and if one is real, both eigenvalues are real. The eigenvalues, denoted by $\gamma$ and $\delta$, are the roots of the equation \begin{equation} \label{eq2} X^2 -(a+d)X +ad-bc=0. \end{equation} If the eigenvalues are not real, $\gamma = \xi + i \eta$ and $\delta = \xi - i \eta$; therefore, $\gamma \delta= \xi^2 + \eta^2 >0$. since $\gamma\delta=ad-bc$, complex values occur only when $ad-bc>0$. \end{remark} When $A$ in ${\cal M}^-$, we denote by $\gamma$ the negative eigenvalue. \begin{proposicion} \label{(C)} \begin{description} \item[(a)] If $ \sigma(T_p) \cap \sigma (A)$ is not empty, then $A$ is in ${\cal S}(T_p)$. More precisely, let $\lambda$ be in $\sigma (T_p) \cap \sigma (A) $, let $D\in \mathbb{R}^2$ be its corresponding $A$-eigenvector, let $\phi \in D(T_p)$ be its corresponding $T_p$-eigenfunction, then $D\phi$ solves (4.1). Consequently, if $\lambda_1 \in \sigma(A)$, and either $b(\lambda_1 -a)>0, \ (\geq 0)$ or $c(\lambda_1 -d)>0, \ (\geq 0)$ the eigenvalue problem $(4.1)$ has a positive (nonnegative) solution. \item[(b)] Conversely, if $A \in {\cal M}^- \cap {\cal S}(T_p)$, then $ \sigma (T_p) \cap \sigma (A) $ is not empty. Moreover if $A \in {\cal M}^-$ and if the eigenvalue problem $(4.1)$ has a positive solution, then $ \sigma (T_p) \cap \sigma (A) = \{\lambda_1\}.$ \end{description} \end{proposicion} This proposition can also be stated as follows:\\ (a) If one of the eigenvalues of $A$ is in $\sigma(T_p)$ then there exists a nontrivial solution of (\ref {p-TP}).\\ (b) Conversely, if $A$ has a negative eigenvalue, and if there exists a nontrivial solution of (\ref {p-TP}), then the other eigenvalue of $A$ is in $\sigma (T_p) $. \begin{remark} In part (b) above, if $\sigma(A) := \{\gamma, \delta \}$ and if $\gamma<0$, necessarily $\delta>0$, and we have $\gamma \delta =ad-bc<0$. \end{remark} \paragraph{Proof of Proposition (\ref{(C)})} {\bf (a)} Assume that $\lambda \in \sigma (A) \cap \sigma(T_p)$. By definition of $\lambda$, there exists an eigenfunction $\varphi \in D(T_p), \ \varphi $ such that $T_p \varphi = \lambda \varphi$. Since $\lambda \in \sigma (A) \subset \mathbb{R}$, there exists an eigenvector $D= \left( \begin{array}{l} d_1 \\ d_{2} \end{array} \right) \in \mathbb{R}^2$ such that $AD= \lambda D$. Define $(\eta, \zeta):= (d_1 \varphi, d_{2} \varphi) $. Since $T_p$ is homogeneous of order 1, $$ T_p \left( \begin{array}{c} \eta \\ \zeta \end{array} \right) = T_p D \varphi = D T_p \varphi = \lambda D \varphi = A D \varphi = A \left( \begin{array}{c} \eta \\ \zeta \end{array} \right) $$ i.e. $(\eta, \zeta)$ is a nontrivial solution of (\ref{p-TP}), and $ (d_1 \phi_{p'} (\varphi), d_{2} \phi_{p'}(\varphi)) \neq (0,0)$ is a nontrivial solution of (\ref{p-EP}). Moreover, if $\lambda=\lambda_1$, we can take $\varphi>0$, and either $(w,z)= (|b|,|\lambda_1 -a|)\phi$ or $(w,z)=(|c|,|\lambda_1 -d|)\phi$ is a positive (nonnegative) solution (or $b=c=0, a=d=\lambda_1$ and $(1,0)\phi, \ (0,1)\phi$ are nonnegative solutions). {\bf (b)} Let $(w,z) \neq (0,0)$ be a nontrivial solution of (\ref{p-TP}), i.e. $(w,z)$ is a nontrivial solution of \begin{eqnarray} \label{syst} &T_pw=aw+bz& \nonumber\\ &T_pz=cw+dz& \\ &w=z=0 \quad\mbox{on }\partial \Omega\,.&\nonumber \end{eqnarray} We first consider some obvious cases. If $w=0$, then $z\not = 0$ satisfies $T_p z= d z$, therefore $d\in \sigma(T_p)$ and $(T_p - a I)w= b z$ implies $b=0;$ consequently $d \in \sigma (A)$ is an eigenvalue of $A$ and of $T_p$. Likewise, $z=0$ implies that $a\in \sigma (A)$ is an eigenvalue of $A$. Hence we assume now that $w\neq 0, \ z\neq 0.$ If $b=0$, then $T_p w= a w$ with $w\neq 0$, implies that $a \in \sigma (A) \cap \sigma(T_p)$. On the same way, $c=0$ implies that $d\in \sigma (A) \cap \sigma(T_p)$. Hence we assume now that $bc \neq 0$. Now $bc\neq 0, $ and assume that $\gamma$ is a negative eigenvalue of $A$. Moreover let us assume that $w\neq 0, \ z\neq 0$ are solutions of (\ref{p-TP}). System (\ref{p-TP}) can also be written as \begin{equation} \label{sistemagamma} (T_p -\gamma I) \left( \begin{array}{c} w \\ z \end{array} \right) = \left( \begin{array}{c c} a -\gamma & b \\ c & d -\gamma \end{array} \right) \left( \begin{array}{c} w \\ z \end{array} \right). \end{equation} Moreover, since $\gamma \in \sigma (A)$, it satisfies \begin{equation} \label{determinante} (a - \gamma )(d - \gamma) = bc. \end{equation} From the first equation in (\ref{syst}), we obtain $(T_p - a I)w= b z$ with $z\in D(T_p)$. Applying $(T_p - \gamma I)$ on both sides of this equation, and taking into account the second equation in (\ref{syst}) we obtain: $$ (T_p -\gamma I)(T_p - a I)w = (T_p -\gamma I)bz = bcw + b(d - \gamma) z.$$ Taking into account ( \ref{determinante}) and (\ref{sistemagamma}) we derive \begin{equation} \label{Tp-eigenvalues} (T_p -\gamma I)(T_p - a I)w = (d - \gamma ) [ (a - \gamma) w +bz] = (T_p -\gamma I)(d - \gamma)w . \end{equation} We observe that, in order to obtain the previous relations, we use the homogeneity of $T_p$ but we cannot commute $T_p-\gamma I$ and $T_p-aI$ because of the non linearity of $T_p$. Since $\gamma <0$, $(T_p-\gamma I)^{-1}$ is well defined, applying it into (\ref{Tp-eigenvalues}) we obtain: $(T_p - a I)w =(d - \gamma)w $, or equivalently $$T_p w \, = \, ( a + d - \gamma)w $$ so that $a+d-\gamma$ is an eigenvalue of $T_p$. Since the eigenvalues of $A$ are real and equal to \begin{equation} ( a+d)/2 \pm \sqrt{((a-d)/2)^2+bc}, \end{equation} if $\gamma<0$ is an eigenvalue of $A$, the other is $\delta=a+d-\gamma$. Moreover, if $w>0$, $z>0$, $T_p w \, = \, \delta w $ implies $\delta=\lambda_1$. \section{The Leray-Schauder degree for the eigenvalue problem} In this section we study the Leray-Schauder degree in terms of the Jordan canonical form of matrices $A \in {\cal M}^-$. For this purpose we use the following property: If $A \in {\cal M}^-$ and $\sigma (A) \cap \sigma (T_p) = \emptyset$ then $A \notin {\cal S}(T_p)$. Therefore $(\ref{p-TP})$ has only the trivial solution, which comes from Proposition \ref{(C)}.(b). We denote by $\ \sigma (A ) = \{ \gamma, \delta \} \ $ the spectrum of Matrix $A$ , and $ \ \sigma (-\Delta _p) \ = \ \sigma (T_p) \ $ the set of eigenvalues of the operator $ \ -\Delta_p \ $ with Dirichlet boundary conditions. \begin{proposicion} \label{deg-LS} Let ${\cal U }\subset (L^{\beta (p)}(\Omega))^2$ be open bounded and $0 \in \cal U$. Let $J$ be the Jordan canonical form of the matrix $A $. Assume that $A \in {\cal M}^-$ and $\sigma (A) \cap \sigma (-\Delta_p ) = \emptyset $. Then $$ \deg_{LS} ( I - T_p ^{-1} A, {\cal U} , 0) = \deg_{LS} ( I - T_p^{-1} J, {\cal U }, 0). $$ Moreover, one of the following two conditions is satisfied \begin{enumerate} \item $ J =\left( \begin{array}{c c}\gamma & 0 \\ 0 & \delta \end{array} \right)$ and $\deg_{LS} ( I - T_p^{-1} J, {\cal U} , 0)$\\ $=\deg_{LS} ( I - \gamma T_p^{-1} , {\cal U} \cap L^{\beta (p) }(\Omega), 0) \deg_{LS} ( I - \delta T_p^{-1} , {\cal U} \cap L^{\beta (p) }(\Omega) , 0)$ \item $J =\left( \begin{array}{c c}\gamma & 0 \\ 1 & \gamma\end{array}\right)$ and \\ $ \deg_{LS} ( I - T_p^{-1} J, {\cal U} , 0)= [ \deg_{LS} ( I - \gamma T_p^{-1} , {\cal U }\cap L^{\beta (p) }(\Omega), 0)]^2. $ \end{enumerate} \end{proposicion} \begin{remark} \rm This result has been obtained for $p=2$ in \cite[Proposition 2.1]{F-P}. In this case, the Leray-Schauder degree for compact linear operators applies \cite{De,L-S}). For $p\neq 2$, the question of calculating $\deg_{LS} (I-\gamma T_p^{-1},{\cal U} \cap L^{\beta (p) }(\Omega), 0)$ has been answered in the following cases \begin{itemize} \item When the spatial dimension $N=1$ \cite{G-V,P-E-M}. \item With radial symmetry \cite{An2,P-M}. \item Whenever $\gamma <\lambda_1$ or $\lambda_1<\gamma <\lambda_2$ \cite{An,P-M}). \end{itemize} The other cases are still open problems. We consider here the case $\gamma<0<\lambda_1$. \end{remark} \paragraph{Proof of Proposition \ref{deg-LS}} Let $P$ be the invertible matrix such that $A=P^{-1}JP$. Let $M_{2\times 2}(\mathbb{R})$ be the space of $2\times 2$-matrix with real coefficients. Let us consider a continuous function $ { \cal P } : [0,1] \to M_{2\times 2}(\mathbb{R})$ such that: 1) ${ \cal P } (t)^{-1}$ exists for all $ t \in [0,1]$, 2) ${ \cal P } (0) =I$, and 3) ${ \cal P } (1) = P$. Let us now define the homotopy $ h:[0,1] \times ( L^{\beta (p) }(\Omega))^2 \to (L^{\alpha (p) }(\Omega))^2 $ by $$ h \left( t, \left( \begin{array}{c} w \\ z \end{array} \right) \right) = T_p^{-1} \left[ { \cal P } (t)^{-1}J { \cal P } (t)\right] \left( \begin{array}{c} w \\ z \end{array} \right), $$ so that $$ h \left( 0, \left( \begin{array}{c} w \\ z \end{array} \right) \right) = T_p^{-1} J \left( \begin{array}{c} w \\ z \end{array} \right), \quad h \left( 1, \left( \begin{array}{c} w \\ z \end{array} \right) \right) = T_p^{-1}A \left( \begin{array}{c} w \\ z \end{array} \right) $$ If there exists some nontrivial solution of $$ h \left( t, \left( \begin{array}{c} w \\ z \end{array} \right) \right) = \left( \begin{array}{c} w \\ z , \end{array} \right) $$ then $\left[ { \cal P } (t)^{-1}J { \cal P } (t)\right] \in {\cal S}(T_p)$ which is impossible by Proposition \ref{(C)}, since $ \sigma \left[ { \cal P } (t)^{-1}J { \cal P } (t)\right] =\sigma(J) = \sigma(A)$, $A\in {\cal M}^-$ and $\sigma(T_p) \cap \sigma \left[ { \cal P } (t)^{-1}J { \cal P } (t)\right] = \emptyset$. So $ h \big( t, \big( \begin{array}{c} w \\ z \end{array} \big) \big) \neq \big( \begin{array}{c} w \\ z \end{array} \big) $ for any $ \big( \begin{array}{c} w \\ z \end{array} \big) \neq \big( \begin{array}{c} 0 \\ 0 \end{array} \big)$. Now the invariance property for homotopies of the Leray-Schauder degree proves that $$ \deg_{LS} ( I - T_p^{-1} A, { \cal U } , 0) = \deg_{LS} ( I - T_p^{-1} J, { \cal U } , 0), $$ with $A=P^{-1}JP.$ We consider separately the following two cases:\\ {\bf Case $(i)$:} By the product formulae \cite[Theorem 8.5]{De}, and since $\gamma, \delta \notin \sigma (-\Delta_p)$, we have $$ \deg_{LS} ( I-T_p^{-1} J, { \cal U } , 0) = \deg_{LS} ( (I - \gamma T_p^{-1},I) , { \cal U } , 0) \deg_{LS} ( (I, I - \delta T_p^{-1}) , K , 0) $$ where $(I - \gamma T_p^{-1},I) (w,z) = ((I - \gamma T_p^{-1})w,z) $ and $K$ is the connected component of $ \ L^{\beta (p) }(\Omega)^2 \setminus ( I - T_p^{-1} J)(\partial { \cal U } )$ containing zero. The reduction property states that $$ \deg_{LS} ( (I - \gamma T_p^{-1},I) , { \cal U } , 0) =\deg_{LS} ( I - \gamma T_p^{-1} , { \cal U } \cap L^{\beta (p) }(\Omega), 0) $$ and Part $(i)$ is proved. \\ {\bf Case (ii):} $ J = \left( \begin{array}{c c}\gamma & 0 \\ 1 & \gamma \end{array} \right)$. Here $ \sigma (A) = \{ \gamma, \, \gamma<0 \} $ and $A$ is a non-diagonalizable matrix. Let us define the homotopy $H:[0,1] \times ( L^{\beta (p)}(\Omega))^2 \to ( L^{\alpha (p)}(\Omega))^2 $ by $$ H \left( t, \left( \begin{array}{c} w \\ z \end{array} \right) \right) = (T_p)^{-1} \left( \begin{array}{c c}\gamma & 0 \\ t & \gamma \end{array} \right) \left( \begin{array}{c} w \\ z \end{array} \right). $$ We have $\sigma \left( \begin{array}{c c}\gamma & 0 \\ t & \gamma \end{array} \right) = \sigma \left( \begin{array}{c c}\gamma & 0 \\ 0 & \gamma \end{array} \right) = \sigma (A) $. By Proposition \ref{(C)}, and due to $A\in {\cal M}^-$, if $H(t,.)$ has a non-trivial solution then $ \sigma \left( \begin{array}{c c}\gamma & 0 \\ t & \gamma \end{array} \right) \cap \sigma (-\Delta _p ) \neq \emptyset,$ which contradicts the hypothesis $ \sigma (A) \cap \sigma (-\Delta _p) = \emptyset . $ Therefore $ \deg_{LS} ( I - H(t,.), {\cal U } , 0) $ is well defined and independent of $t\in [0,1]$. Moreover by using again the product formulae \begin{eqnarray*} \deg_{LS} ( I - (T_p)^{-1} J, {\cal U} , 0) & = & \deg_{LS} ( I - H(1,.), {\cal U }, 0) \\ & = & \deg_{LS} (I - H(0,.), {\cal U }, 0) \\ & = & [\deg_{LS} ( I - \gamma (T_p)^{-1} , {\cal U }\cap L^{\beta (p)}(\Omega) , 0)]^2. \end{eqnarray*} \section{Existence of Positive Bifurcated Solutions} In this section we study sufficient conditions for the existence of positive solutions bifurcating from $(A_0 , (0,0) )$ where $A_0= \left( \begin{array}{cc} a_0 & b_0 \\ c_0 & d_0 \end{array} \right)$. From Theorem~\ref{bif.p-EP} we will need that the eigenvalue problem has a (nontrivial) non negative solution, and therefore we will require, from Proposition \ref{(C)}, that $\lambda_1 \in \sigma ( A_0)$ and therefore $\sigma(T_p) \cap \sigma ( A_0) \not= \emptyset$. Another usual requirement is that there is a changement of topological degree (cf. \cite{Ra},\cite[Theorem 28.1]{De}, \cite{A-A}, ...). More explicitly, we have the following \begin{theorem} Assume that $f,g$ satisfy (H), that $\lambda_1 \in \sigma ( A_0)$ and that $A_0 \in \cal M.$ Then $(A_0 , (0,0) )$ is a bifurcation point to positive solutions of (\ref{p-System}) in $\mathbb{R}^4 \times (W_0^{1,p}(\Omega) )^2$. Moreover, there is a connected component of topological dimension $\geq 4$ of the set of nontrivial solutions of (\ref{p-System}) in $\mathbb{R}^4 \times (W_0^{1,p}(\Omega) )^2$ whose closure contains the point $(A_0, (0,0))$. \end{theorem} \begin{remark} \rm Theorem above is a generalization for systems of the already known situation for one single equation \cite[Proposition 2.2]{P-M}. \end{remark} \paragraph{Proof.} Hereafter we denote by $B_E(c,r)$ the ball in some space $E$ with center $c \in E$ and radius $r$. Suppose that $(A_0 , (0,0) )$ is not a bifurcation point of (\ref{p-System}). Since $\lambda_1$ is isolated, there are $\epsilon_0 >0$ and $r_0>0$ such that for every $A \in B_{\mathbb{R}^4}( A_0, \epsilon_0) \subset \mathbb{R}^4 \quad $, if $(w,z) \in B_{(W_0^{1,p})^2}((0,0), r_0) \subset (W_0^{1,p})^2$ satisfies (\ref{p-System}), then $(w,z)=(0,0)$. Since for any $A \in B_{\mathbb{R}^4}( A_0 , \epsilon_0)$ the functions $$ f(a , .,.) , \ g(d ,.,.) : (L^{\alpha})^2 \to L^{\beta } $$ map bounded sets into bounded sets, the function $ F: B_{\mathbb{R}^4}( A_0 , \epsilon_0 ) \times (L^{\alpha})^2 \to (L^{\alpha})^2$ given by $$ F \left( A , \left( \begin{array}{c} w \\ z \end{array} \right) \right) = T_p^{-1} \left[ \left( \begin{array}{c c}a & b \\ c & d \end{array} \right) \left( \begin{array}{c} w \\ z \end{array} \right) + \left( \begin{array}{c} f(a , w,z) \\ g(d ,w,z) \end{array} \right) \right] $$ is completely continuous, consequently $ \deg_{LS} ( I - F(A,.), B_{(L^{\alpha})^2} ((0,0), r_0),0)$ is well defined and independent of $A\in B_{\mathbb{R}^4}( A_0 , \epsilon_0 )$. % For $A_0= \left( \begin{array}{c c} a_0 & b_0 \\ c_0 & d_0 \end{array} \right)$, denote by $J_0$ its Jordan canonical form. By hypothesis we can always choose two matrices, $A_i= \left( \begin{array}{c c}a_i & b_i \\ c_i & d_i \end{array} \right)$, $i=1,2$, such that \begin{enumerate} \item[(a)] $\sigma (A_i) \cap \sigma (-\Delta_p) = \emptyset ,$ \item[(b)] $A_i \in \cal M$ \item[(c)] $ A_i \in B_{\mathbb{R}^4}( A_0 , \epsilon_0 ) $ and \item[(d)] $ \deg_{LS} ( I - T_p ^{-1} A_1, { \cal U } , 0) \neq \deg_{LS} ( I - T_p^{-1} A_{2}, { \cal U } , 0).$ \end{enumerate} Let us now define the homotopies $$ H_i \left( t, \left( \begin{array}{c} w \\ z \end{array} \right) \right) = T_p^{-1} \left[ A_i \left( \begin{array}{c} w \\ z \end{array} \right) + t \left( \begin{array}{c} f(a , w,z) \\ g(d ,w,z) \end{array} \right) \right], $$ Next, we show by contradiction that there exists a real number sufficiently small again denoted by $r_0$ such that $$ H_i \left( t, \left( \begin{array}{c} w \\ z \end{array} \right) \right) \neq \left( \begin{array}{c} w \\ z \end{array} \right) \quad \mbox{in} \quad \partial B_{(L^{\alpha})^2}((0,0), r_0) \subset (L^{\alpha})^2 $$ for any $t\in [0,1]$. Assume that for any $n\in \mathbb{N}$ large enough, there exists a sequence $$ \left\{ \left( t_{n} , \left( \begin{array}{c} w_{n} \\ z_{n} \end{array} \right) \right) \right\} \in [0,1] \times (L^{\beta})^2, \ \ \ \left\| \left( \begin{array}{c} w_{n} \\ z_{n} \end{array} \right) \right\|_{(L^{\beta})^2} = 1/n , $$ and $$ T_p \left( \begin{array}{c} w_{n} \\ z_{n} \end{array} \right) = A _i \left( \begin{array}{c} w_{n} \\ z_{n} \end{array} \right) + t_{n} \left( \begin{array}{c} f(a , w_{n},z_{n}) \\ g(d ,w_{n},z_{n}) \end{array} \right) , \quad\mbox{in }\Omega, $$ $$ w_{n}=z_{n}=0, \quad\mbox{on }\partial \Omega. $$ Arguing as in the proof of Theorem (\ref{bif.p-EP}), the associated eigenvalue problem has a non-trivial solution which is positive. Hence, by Proposition (\ref{(C)}), $$\lambda_1 \in \sigma (A_i) \cap \sigma (-\Delta_p) $$ which contradicts (a). Then $H_i (t, (w,z)) \neq (w,z) $ in $ \partial B_{(L^{\alpha})^2}((0,0), r_0 ) $, therefore $ \deg_{LS} ( I - H_i(t,.), B_{(L^{\alpha})^2} ((0,0), r_0),0)$ is well defined and independent of $t$, consequently \begin{eqnarray*} \lefteqn{ \deg_{LS} ( I - F(A_i , . ), B_{(L^{\alpha})^2} ((0,0), r_0),0)}\\ &=&\deg_{LS} ( I - H_i(1,.), B_{(L^{\alpha})^2} ((0,0), r_0),0) \\ & = & \deg_{LS} ( I - H_i(0,.), B_{(L^{\alpha})} ((0,0), r_0),0) \\ & = & \deg_{LS} ( I - T_p ^{-1} A_i, B_{(L^{\alpha})^2} ((0,0), r_0) , 0) \end{eqnarray*} which, jointly with (c) and (d), contradicts the assertion that \\ $ \deg_{LS} ( I - F(A_i ,.), B_{(L^{\alpha})^2} ((0,0), r_0),0)$ is constant for $ A\in B_{\mathbb{R}^4}( A_0 , \epsilon_0 ) $. Now, we built the nonnegative the matrices $A_1, A_2$. >From the definition of the Jordan's canonical form, there exists an invertible matrix $P$ such that $A_0=P^{-1}J_0P$. Denote now by $\gamma<0$ and $\delta$ the eigenvalues of $A$. Assume that $\delta = \lambda_1$. 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Optim., {\bf 12}, 191-202, (1984). \end{thebibliography} \noindent{\sc Jacqueline Fleckinger} \\ {\sc CEREMATH \& UMR MIP}, Universit\'{e} Toulouse 1 \\ pl. A. France\\ 31042 Toulouse Cedex, France \\ e-mail: jfleck@univ-tlse1.fr \smallskip \noindent{\sc Rosa Pardo} \\ Departamento de Matem\'atica Aplicada \\ Universidad Complutense de Madrid \\ Madrid 28040, Spain \\ e-mail: rpardo@sunma4.mat.ucm.es \smallskip \noindent{\sc Fran\c{c}ois de Th\'elin}\\ {\sc UMR MIP}, Universit\'{e} Toulouse 3 \\ 118 route de Narbonne\\ 31062 Toulouse Cedex 04, France \\ e-mail: dethelin@mip.ups-tlse.fr \end{document}