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.
Categories: math.AP