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.