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

Published online by Cambridge University Press:  03 March 2014

## Abstract

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

$$(P_{\var})\hspace*{4cm} \left\{ \begin{array}{rcl} \mathcal{L}_{\var}u=f(u) \ \ \mbox{in} \ \ \R^{3},\\[1.5mm] u>0 \ \ \mbox{in} \ \ \R^{3},\\[1.5mm] u \in H^{1}(\R^3), \end{array} \right.$$ $\mathrm{\left(}{\mathit{P}}_{\mathit{\epsilon }}\mathrm{\right)}\left\{\begin{array}{c}\\ {\mathrm{ℒ}}_{\mathit{\epsilon }}\mathit{u}\mathrm{=}\mathit{f}\mathrm{\left(}\mathit{u}\mathrm{\right)}\mathrm{in}{\mathrm{IR}}^{\mathrm{3}}\mathit{,}\\ \mathit{u}\mathit{>}\mathrm{0}\mathrm{in}{\mathrm{IR}}^{\mathrm{3}}\mathit{,}\\ \mathit{u}\mathrm{\in }{\mathit{H}}^{\mathrm{1}}\mathrm{\left(}{\mathrm{IR}}^{\mathrm{3}}\mathrm{\right)}\mathit{,}\end{array}\right\$

where ε is a small positive parameter, f : ℝ → ℝ is a continuous function, $$\mathcal{L}_{\var}$$ ${\mathrm{ℒ}}_{\mathit{\epsilon }}$ is a nonlocal operator defined by

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

M : IR+ → IR+ and V : IR3 → IR are continuous functions which verify some hypotheses.

## Keywords

Type
Research Article
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Copyright
© EDP Sciences, SMAI, 2014

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