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Un résultat de bifurcation globale à partir de la première valeur propre pour un système de \(p\)-laplaciens. (A result of global bifurcation from the first eigenvalue for a system of \(p\)-Laplacians). (French) Zbl 0777.35009

Summary: We study the following system, defined on a regular bounded open set \(\Omega\subset\mathbb{R}^ N\) for \(p>1\): \[ -\Delta_ p u=\widehat\lambda\phi_ p(u)+b\phi_ p(v)+f(\widehat\lambda,\phi_ p(u)),\quad -\Delta_ p v=c\phi_ p(u)+d\phi_ p(v)\quad\text{in }\Omega,\tag\text{Sp} \]
\[ u=v=0\quad\text{on }\partial\Omega.\tag\text{BC} \] Here \(\phi_ p(u):=| u|^{p- 2}u\) and \(\Delta_ p:=\text{Div}(|\nabla u|^{p-2}\nabla u)\) is the \(p\)-Laplacian; \(\widehat\lambda\) is a real parameter and \(b\), \(c\) and \(d\) are given numbers; \(\lambda_ 1(p)\) is the first eigenvalue of \(- \Delta_ p\) defined on \(\Omega\) with Dirichlet boundary conditions.
A solution of (Sp)-(BC) is a pair \((u,v)\in W^{1,p}_ 0(\Omega)\times W^{1,p}_ 0(\Omega)\) satisfying (Sp) in the weak sense. We also study the associated eigenvalue problem (Ep)-(BC) with: \[ -\Delta_ p u=\widehat\lambda\phi_ p(u)+b\phi_ p(v),\quad-\Delta_ p v=c\phi_ p(u)+d\phi_ p(v)\quad\text{in }\Omega;\tag\text{Ep} \] if \((u,v)\in W^{1,p}_ 0(\Omega)\times W^{1,p}_ 0(\Omega)\) is a nonzero solution of (Ep)-(BC) for some \(\widehat\lambda\), then \(\widehat\lambda\) is called an eigenvalues of the problem. We show that \(\widehat\lambda_ 1(p):=\lambda_ 1(p)-bc(\lambda_ 1(p)-d)^{-1}\) is the smallest eigenvalue of (Ep)-(BC); it is isolated and is associated with a unique positive (up to constant factor) eigenfunction.
Theorem: Let \((\widehat\mu,(0,0))\) be a bifurcation point of (Sp)-(BC) in \(\mathbb{R}\times (W^{1,p}_ 0(\Omega))^ 2\), then \(\widehat\mu\) is an eigenvalue of (Ep)-(BC). Conversely, \((\widehat\lambda_ 1(p),\;(0,0))\) is a bifurcation point of (Sp)-(BC) in \(\mathbb{R}\times (W^{1,p}_ 0(\Omega))^ 2\).

MSC:

35B32 Bifurcations in context of PDEs
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35J70 Degenerate elliptic equations
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