Existence and concentration of positive solutions for a quasilinear elliptic equation in R - Elisandra Gloss (Electronic Journal of Differential Equations, 2010)
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 $mathbbR$ as $varepsilono 0$, where the potential $V:mathbbRo mathbbR$ has a positive infimum and $inf_{partial Omega}V>inf_{ Omega}V$ for some bounded domain $Omega$ in $mathbbR$, and $h$ is a nonlinearity without having growth conditions such as Ambrosetti-Rabinowitz.