Multiplicity and concentration behavior of positive solutions for a Schrodinger-Kirchhoff type problem via penalization method

Abstract

In this paper we are concerned with questions of multiplicity and concentration behavior of positive solutions of the elliptic problem $$\left\{\begin{array}{rcl} \mathcal{L}_{\varepsilon}u = f(u) \ \ \mbox{in} \ \ \mathbb{R}^3,\\ u>0 \ \ \mbox{in} \ \ \mathbb{R}^3,\\ u \in H^1 (\mathbb{R}^3), \end{array} \right.$$ where $\epsilon$ is a small positive parameter, $f:\mathbb{R}\to \mathbb{R}$ is a continuous function, ${\mathcal{L}}_{\epsilon }$ is a nonlocal operator defined by $${\mathcal{L}}_{\epsilon }u=M\left(\frac{1}{\epsilon }{\int }_{{\mathbb{R}}^3}|\nabla u{|}^2+\frac{1}{{\epsilon }^3}{\int }_{{\mathbb{R}}^3}V(x)u^2\right)\left[-{\epsilon }^2\Delta u+V(x)u\right],$$ $M:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ and $V:{\mathbb{R}}^3\to \mathbb{R}$ are continuous functions which verify some hypotheses.

32 pages

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