# 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 ellipticÂ 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}.

Published

2013-08-24

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Articles