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In this paper we are concerned with questions of multiplicity andconcentration 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 functionswhich verify some hypotheses.

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