# Three solutions for pertubed Dirichilet problem

Giuseppe Rao, Giuseppe Cordaro

Risultato della ricerca: Articlepeer review

## Abstract

\begin{abstract}In this paper we prove the existence of at least three distinct solutionsto the following perturbed Dirichlet problem\begin{displaymath}\left\{\begin{array}{ll}-\Delta u= f(x,u)+\lambda g(x,u) & \mbox{in\ } \Omega\\u=0 & \mbox{on\ } \partial \Omega,\end{array}\right.\end{displaymath}where $\Omega\subset\mathbb{R}^N$ is an open bounded set with smoothboundary $\partial \Omega$ and $\lambda\in \mathbb{R}$. Under very mildconditions on $g$ and some assumptions on the behaviour of the potentialof $f$ at $0$ and $+\infty$, our result assures the existence of at leastthree distinct solutions to the above problem for $\lambda$ small enough.Moreover such solutions belong to a ball of the space $W_0^{1,2}(\Omega)$centered in the origin and with radius not dependent on $\lambda$.\end{abstract}
Lingua originale English 3879-3883 NONLINEAR ANALYSIS 68 Published - 2008

## All Science Journal Classification (ASJC) codes

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