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

Figueiredo, Giovany M.; Santos, João R.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 2, pp. 389-415.

In this paper we are concerned with questions of multiplicity and concentration behavior of positive solutions of the elliptic problem

$$\left(P_{\epsilon }\right)\left\{\begin{array}{c}\hfill {ℒ}_{\epsilon }u=f\left(u\right)\text{in}{ℝ}^3,\\ \hfill u>0\text{in}{ℝ}^3,\\ \hfill u\in H^1\left({ℝ}^3\right),\end{array}\right.$$

where $\epsilon$ is a small positive parameter, $f:ℝ\to ℝ$ is a continuous function, ${ℒ}_{\epsilon }$ is a nonlocal operator defined by

$${ℒ}_{\epsilon }u=M\left(\frac{1}{\epsilon }{\int }_{{ℝ}^3}{\left|\nabla u\right|}^2+\frac{1}{{\epsilon }^3}{\int }_{{ℝ}^3}V\left(x\right)u^2\right)\left[{-}^2\Delta u+V\left(x\right)u\right],$$

$M:{ℝ}_{+}\to {ℝ}_{+}$ and $V:{ℝ}^3\to ℝ$ are continuous functions which verify some hypotheses.

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