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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

Giovany M. Figueiredo
Affiliation:
Faculdade de Matemática, Universidade Federal do Pará, 66.075-110 Belém, Pará, Brazil
João R. Santos Júnior
Affiliation:
Faculdade de Matemática, Universidade Federal do Pará, 66.075-110 Belém, Pará, Brazil
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Abstract

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

$$ (P_{\var})\hspace*{4cm} \left\{ \begin{array}{rcl} \mathcal{L}_{\var}u=f(u) \ \ \mbox{in} \ \ \R^{3},\\[1.5mm] u>0 \ \ \mbox{in} \ \ \R^{3},\\[1.5mm] u \in H^{1}(\R^3), \end{array} \right. $$(Pε)εu=f(u)inIR3,u>0inIR3,uH1(IR3),

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

$$ \mathcal{L}_{\var}u=M\left(\dis\frac{1}{\var}\int_{\R^{3}}|\nabla u|^{2}+\frac{1}{\var^{3}}\dis\int_{\R^{3}}V(x)u^{2}\right)\left[-\var^{2}\Delta u + V(x)u \right], $$εu=M1εIR3|u|2+1ε3IR3V(x)u2[ε2Δu+V(x)u],

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

Type
Research Article
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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