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 $\varepsilon$ is a small positive parameter, $f:\mathbb{R}\rightarrow \mathbb{R}$ is a continuous function, $\mathcal{L}{\varepsilon}$ is a nonlocal operator defined by $$ \mathcal{L}{\varepsilon} u = M \left(\frac{1}{\varepsilon} \int{\mathbb{R}^3} |\nabla u|^2 + \frac{1}{\varepsilon^3} \int_{\mathbb{R}^3} V(x) u^{2}\right) \left[-\varepsilon^2\Delta u + V(x)u \right],$$ $M:\mathbb{R}+\to \mathbb{R}+$ and $V:\mathbb{R}^3 \to \mathbb{R}$ are continuous functions which verify some hypotheses.