Positive solutions of strongly nonlinear elliptic problems

Abstract

Abstract

We study existence of multiple positive solutions for the nonlinear eigenvalue problem $-div\mathrm{(}\mathit{\varphi }\mathrm{(}|\nabla \mathit{u}|\mathrm{)}\nabla \mathit{u}\mathrm{)}=\mathit{\lambda }\mathit{f}\mathrm{(}\mathit{u}\mathrm{)}$ in Ω, $\mathit{u}=0$ on $\mathit{\partial }\mathit{\Omega }$ , where $\mathit{\Omega }\subset {\mathbf{R}}^{\mathit{N}}$ is a bounded domain with smooth boundary $\mathit{\partial }\mathit{\Omega }$ , $\mathit{\varphi }:\mathrm{(}0\mathrm{,}\infty \mathrm{)}\to \mathrm{(}0\mathrm{,}\infty \mathrm{)}$ is a suitable ${\mathit{C}}^1$ -function, $\mathit{\lambda }\mathrm{>}0$ is a parameter and $\mathit{f}:[0\mathrm{,}\infty \mathrm{)}\to \mathbf{R}$ is a sign-changing continuous function. We show existence of a finite number of solutions in the case f changes sign a finite number of times and existence of infinitely many solutions in the case f changes sign an infinite number of times. We employ variational arguments, regularity results, a strong maximum principle by Pucci and Serrin and a general result on lower and upper solutions. Our research was motivated by the work of Hess for the case of the Laplacian and Loc and Schmitt for the case of the p-Laplacian and we were able to extend the major results by Loc and Schmitt.