Francisco
O. V. De Paiva and Eugenio
Massa*, Multiple Solutions for Some Elliptic Equations With a
Nonlinearity Concave in the Origin*

Abstract

In this paper we establish the existence of multiple solutions for the semilinear elliptic problem

\[

\begin{array}{lll}

-\Delta u = -\lambda |u|^{q-2}u +au+g(u) & {\rm in} & \Omega\\

\ \ \ \ u = 0 & {\rm on} & \partial \Omega,

\end{array}

\]

where $\Omega \subset \mathbb{R}^N$ is a bounded domain with smooth boundary $\partial \Omega$, $g:\mathbb{R}\to \mathbb{R}$ is a function of class $C^1$ such that $g(0)=g'(0)=0$, $\lambda>0$ is real parameter, $a\in\R$, and $1<q<2$. We study the problem when $g$ is superlinear, asymptotically linear and asymmetric or infinity.

Key words and phrases: multiplicity of solution

1991 Mathematical Subject Classification: 35J65 (35J20)

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the authors either via e-mail or by snail mail, at the address:

IMECC, UNICAMP

Cx. P. 6065

13083-970 Campinas, SP, BRAZIL

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indíce de Relatórios de Pesquisa**

In this paper we establish the existence of multiple solutions for the semilinear elliptic problem

\[

\begin{array}{lll}

-\Delta u = -\lambda |u|^{q-2}u +au+g(u) & {\rm in} & \Omega\\

\ \ \ \ u = 0 & {\rm on} & \partial \Omega,

\end{array}

\]

where $\Omega \subset \mathbb{R}^N$ is a bounded domain with smooth boundary $\partial \Omega$, $g:\mathbb{R}\to \mathbb{R}$ is a function of class $C^1$ such that $g(0)=g'(0)=0$, $\lambda>0$ is real parameter, $a\in\R$, and $1<q<2$. We study the problem when $g$ is superlinear, asymptotically linear and asymmetric or infinity.

Key words and phrases: multiplicity of solution

1991 Mathematical Subject Classification: 35J65 (35J20)

IMECC, UNICAMP

Cx. P. 6065

13083-970 Campinas, SP, BRAZIL

November 25, 2004