Existence of multiple solutions for a Dirichlet boundary value problem driven by a \emph{p}-Laplacian operator

Abstract

In this paper we prove the existence of at least three distinct solutions to the following perturbed Dirichlet problem:
\begin{equation*}
\begin{cases}
\nabla (\abs{\nabla u}^{p-2} \nabla u) = f(x,u) + \lambda g(x,u) & \text{in }\Omega\\
u= 0 & \text{on } \partial \Omega,
\end{cases}
\end{equation*}
where $\Omega \subset \RR^N$ is an open bounded set with smooth boundary $\partial \Omega$ and $\lambda \in \RR$. Under very mild conditions on $g$ and some assumptions on the behaviour of the potential of $f$ at 0 and $+ \infty$, our result assures the existence of at least three distinct solutions to the above problem for $\lambda$ small enough. Moreover such solutions belong to a ball of the space $W_0^{1,p}(\Omega)$ centered in the origin and with radius not dependent on $\lambda$.