Electron. J. Diff. Eqns., Vol. 2001(2001), No. 06, pp. 1-15.

Four-parameter bifurcation for a p-Laplacian system

Jacqueline Fleckinger, Rosa Pardo, & Francois de Thelin

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 $p greater than 1$ and $p \neq 2$, is the $p$-Laplacian operator, and $\phi_{p} (s) =|s|^{p-2} s$ with $p greater than 1$. We assume that $a, b, c, d$ are real parameters. Then we use a bifurcation method to exhibit some nontrivial solutions. The associated eigenvalue problem, with $f=g \equiv 0$, is also studied here.

Submitted June 29, 2000. Published January 9, 2001.
Math Subject Classifications: 35J45, 35J55, 35J60, 35J65, 35J30, 35P30.
Key Words: p-Laplacian, bifurcation.

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Jacqueline Fleckinger
CEREMATH & UMR MIP, Universite Toulouse 1
pl. A. France
31042 Toulouse Cedex, France
e-mail: jfleck@univ-tlse1.fr
Rosa Pardo
Departamento de Matematica Aplicada
Universidad Complutense de Madrid
Madrid 28040, Spain
e-mail: rpardo@sunma4.mat.ucm.es
Francois de Thelin
UMR MIP, Universite Toulouse 3
118 route de Narbonne
31062 Toulouse Cedex 04, France
e-mail: dethelin@mip.ups-tlse.fr

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