Infinitely many positive solutions for the Neumann problem involving the $p$-Laplacian

Giovanni Anello¹ ; Giuseppe Cordaro¹

¹ Department of Mathematics University of Messina 98166 Sant'Agata-Messina, Italy

Colloquium Mathematicum, Tome 97 (2003) no. 2, pp. 221-231.

Résumé

We present two results on existence of infinitely many positive solutions to the Neumann problem $$ \cases{ -{\mit\Delta}_p u+\lambda(x)|u|^{p-2}u = \mu f(x,u) {\rm in}\ {\mit\Omega},\cr \partial u/\partial \nu=0 {\rm on}\ \partial{\mit\Omega},\cr} $$ where ${\mit\Omega} \subset {\mathbb R}^N$ is a bounded open set with sufficiently smooth boundary $\partial {\mit\Omega}$, $\nu$ is the outer unit normal vector to $\partial {\mit\Omega}$, $p>1$, $\mu>0$, $\lambda\in L^\infty({\mit\Omega})$ with $\mathop{\rm ess\,inf}_{x\in{\mit\Omega}}\lambda(x)>0$ and $f:{\mit\Omega}\times{\mathbb R}\rightarrow{\mathbb R}$ is a Carathéodory function. Our results ensure the existence of a sequence of nonzero and nonnegative weak solutions to the above problem.

Détail

DOI : 10.4064/cm97-2-8

Keywords: present results existence infinitely many positive solutions neumann problem cases mit delta lambda p mit omega partial partial partial mit omega where mit omega subset mathbb bounded set sufficiently smooth boundary partial mit omega outer unit normal vector partial mit omega lambda infty mit omega mathop ess inf mit omega lambda mit omega times mathbb rightarrow mathbb carath odory function results ensure existence sequence nonzero nonnegative weak solutions above problem