Multiple solutions for a class of quasilinear problems

Francisco Odair de Paiva

doi:10.3934/dcds.2006.15.669

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2006, Volume 15, Issue 2: 669-680. Doi: 10.3934/dcds.2006.15.669

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Multiple solutions for a class of quasilinear problems

Francisco Odair de Paiva^1,

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

Received: February 28, 2005

Revised: September 30, 2005

Published: March 2006

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Abstract

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.

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Mathematics Subject Classification: 35J25 (58E05).

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