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

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 $varepsilono 0$, where the potential $V:mathbb{R}o mathbb{R}$ 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.

Keywords

• Schrodinger equation
• quasilinear equation
• concentration
• variational methods