Electron. J. Diff. Eqns., Vol. 2002(2002), No. 25, pp. 1-15.

Existence and multiplicity results for nonlinear elliptic problems in $\mathbb{R}^N$ with an indefinite functional

David G. Costa, Yuxia Guo, & Miguel Ramos

Abstract:
We prove the existence of a nontrivial solution for the nonlinear elliptic problem
$-\Delta u=\lambda h(x)u + a(x)g(u)$ in $\mathbb{R}^N$
where $g$ is superlinear near zero and near infinity, $a(x)$ changes sign, $\lambda $ is positive, and $h(x)\geq 0$ is a weight function. For $g$ odd, we prove the existence of an infinite number of solutions.

Submitted June 23, 2001. Published March 4, 2002.
Math Subject Classifications: 35J25, 35J20, 58E05.
Key Words: Superlinear elliptic problems, Morse index, minimax methods.

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David G. Costa
Dept. of Math. Sciences, University of Nevada,
Las Vegas, NV 89154-4020, USA
e-mail: costa@unlv.edu

Yuxia Guo
Chinese Academy of Sciences
Beijing 100080, China
e-mail: yxguo@lsc02.iss.ac.cn

Miguel Ramos
CMAF, Univ. de Lisboa
Av. Prof. Gama Pinto, 1649-003 Lisboa, Portugal
e-mail: mramos@lmc.fc.ul.pt

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