Multiple solutions for a class of elliptic equations of $p$-Laplacian type with combined nonlinearities

Abstract

In this article, we deal with the mutliplicity of solutions to a class of elliptic  equations of $p$-Laplacian type
$$\begin{cases}
\begin{array}{rlll}
- \operatorname{div}(a(x,\nabla u)) &= & \lambda b(x)f(u) + \mu g(x,u)& \text{ in } \Omega,\\
u &= & 0 & \text{ on } \partial\Omega,
\end{array}
\end{cases}$$
where $\Omega \subset \R^N$, $N \geq 2$ is a bounded domain with smooth boundary $\partial\Omega$, $\lambda, \mu$ are parameters. The function $f: \R \to \R$ is assumed to be $(p-1)$-sublinear at infinity while the function $g:\Omega\times\R\to \R$ is $(p-1)$-asymptotically
linear at infinity with respect to the second variable. The proofs rely essentially on the minimax principle by B. Ricceri \cite{Ricceri} combined with the mountain pass theorem by P. Pucci and J. Serrin \cite{PucSer}.