Electron. J. Diff. Equ., Vol. 2010(2010), No. 61, pp. 1-23.

Existence and concentration of positive solutions for a quasilinear elliptic equation in R

Elisandra Gloss

Abstract:
We study the existence and concentration of positive solutions for the quasilinear elliptic equation
$$ -\varepsilon^2u'' -\varepsilon^2(u^2)''u+V(x) u = h(u) $$
in $\mathbb{R}$ as $\varepsilon\to 0$, where the potential $V:\mathbb{R}\to \mathbb{R}$ has a positive infimum and $\inf_{\partial \Omega}V>\inf_{ \Omega}V$ for some bounded domain $\Omega$ in $\mathbb{R}$, and $h$ is a nonlinearity without having growth conditions such as Ambrosetti-Rabinowitz.

Submitted January 16, 2010. Published May 5, 2010.
Math Subject Classifications: 35J20, 35J62.
Key Words: Schrodinger equation; quasilinear equation; concentration; variational methods.

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Elisandra Gloss
Departamento de Matemática
Universidade Federal da Paraíba
58000-000, João Pessoa - PB, Brazil
email: elisandra@mat.ufpb.br

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