# American Institute of Mathematical Sciences

May  2006, 15(2): 669-680. doi: 10.3934/dcds.2006.15.669

## Multiple solutions for a class of quasilinear problems

 1 Departamento de Matemática, IMECC - UNICAMP , Caixa Postal 6065, 13081-970 Campinas-SP, Brazil

Received  March 2005 Revised  October 2005 Published  March 2006

In this paper we establish the existence of positive and multiple solutions for the quasilinear elliptic problem

$-\Delta_p u = g(x,u)$  in  $\Omega$
$u = 0$  on  $\partial \Omega$,

where $\Omega \subset \mathbb{R}^N$ is an open bounded domain with smooth boundary $\partial \Omega$, $g:\Omega\times\mathbb{R}\to \mathbb{R}$ is a Carathéodory function such that $g(x,0)=0$ and which is asymptotically linear. We suppose that $g(x,t)/t$ tends to an $L^r$-function, $r>N/p$ if 1 < p ≤ N and $r=1$ if $p>N$, which can change sign. We consider both the resonant and the nonresonant cases.

Citation: Francisco Odair de Paiva. Multiple solutions for a class of quasilinear problems. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 669-680. doi: 10.3934/dcds.2006.15.669
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