Multiple nonnegative solutions for elliptic boundary value problems involving the $p$-Laplacian

Abstract

In this paper we present a result concerning the existence of two nonzero nonnegative solutions for the following Dirichlet problem involving the $p$-Laplacian $$ \cases -\Delta_p u=\lambda f(x,u) &\text{\rm in\ } \Omega,\\ u=0 &\text{\rm on\ } \partial \Omega, \endcases $$ using variational methods. In particular, we will determine an explicit real interval $\Lambda$ for which these solutions exist for every $\lambda\in \Lambda$. We also point out that our result improves and extends to higher dimension a recent multiplicity result for ordinary differential equations.