On Neumann boundary value problems for some quasilinear elliptic equations

Binding, Paul A. ; Drábek, Pavel ; Huang, Yin Xi

Electronic Journal of Differential Equations, Tome 1997 (1997).

Résumé

Summary: We study the role played by the indefinite weight function $a(x)$ on the existence of positive solutions to the problem $$\left\{ \eqalign{ -{\rm div}\,(|\nabla u|^{p-2}\nabla u)\= \lambda a(x)|u|^{p-2}u+b(x)|u|^{\gamma-2}u, \quad x\in\Omega, \cr x{\partial u \over\partial n}\=0, \quad x\in\partial\Omega\,,} \right. $$ where $\Omega$ is a smooth bounded domain in $R^n$ , b changes sign, $1 less than p less than N , 1 less than \gamma less than Np/(N-p)$ and $\gamma\ne p$ . We prove that if $\int_\Omega a(x)\, dx\ne 0$ and b satisfies another integral condition, then there exists some $\lambda^*$ such that $\lambda^* \int_\Omega a(x)\, dx$ and, for $\lambda$ strictly between 0 and $\lambda^*$ , the problem has a positive solution. if $\int_\Omega a(x)\, dx=0$ , then the problem has a positive solution for small $\lambda$ provided that $\int_\Omega b(x)\,dx$ .

Détail

Classification : 35J65, 35J70, 35P30
Keywords: p-Laplacian, positive solutions, Neumann boundary value problems