Infinitely many positive solutions for the Neumann problem involving the $p$-Laplacian
Volume 97 / 2003
Abstract
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.
