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

Giovany M. Figueiredo; João R. Santos

ESAIM: Control, Optimisation and Calculus of Variations (2014)

  • Volume: 20, Issue: 2, page 389-415
  • ISSN: 1292-8119

Abstract

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In this paper we are concerned with questions of multiplicity and concentration behavior of positive solutions of the elliptic problem ( P ) ℒ u = f ( u ) in ℝ 3 , u > 0 in ℝ 3 , u ∈ H 1 ( ℝ 3 ) , ( P ε ) ℒ ε u = f ( u ) in IR 3 , u > 0 in IR 3 , u ∈ H 1 ( IR 3 ) , whereε is a small positive parameter, f : ℝ → ℝ is a continuous function, ℒ ℒ ε is a nonlocal operator defined by ℒ u = M 1 ∫ ℝ 3 | ∇ u | 2 + 1 3 ∫ ℝ 3 V ( x ) u 2 - 2 Δ u + V ( x ) u , ℒ ε u = M 1 ε ∫ IR 3 | ∇ u | 2 + 1 ε 3 ∫ IR 3 V ( x ) u 2 [ − ε 2 Δ u + V ( x ) u ] ,M : IR+ → IR+ and V : IR3 → IR are continuous functions which verify some hypotheses.

How to cite

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Figueiredo, Giovany M., and Santos, João R.. "Multiplicity and concentration behavior of positive solutions for a Schrödinger–Kirchhoff type problem via penalization method." ESAIM: Control, Optimisation and Calculus of Variations 20.2 (2014): 389-415. <http://eudml.org/doc/272888>.

@article{Figueiredo2014,
abstract = {In this paper we are concerned with questions of multiplicity and concentration behavior of positive solutions of the elliptic problem\[ (P\_\{\})\hspace*\{113.81102pt\} \left\lbrace \begin\{array\}\{rcl\} \mathcal \{L\}\_\{\}u=f(u) \ \ \mbox\{in\} \ \ \mathbb \{R\}^\{3\},\\[1.5mm] u&gt;0 \ \ \mbox\{in\} \ \ \mathbb \{R\}^\{3\},\\[1.5mm] u \in H^\{1\}(\mathbb \{R\}^3), \end\{array\} \right. \]( P ε ) ℒ ε u = f ( u ) in IR 3 , u &gt; 0 in IR 3 , u ∈ H 1 ( IR 3 ) , whereε is a small positive parameter, f : ℝ → ℝ is a continuous function,\[ \mathcal \{L\}\_\{\} \]ℒ ε is a nonlocal operator defined by\[ \mathcal \{L\}\_\{\}u=M\left(\frac\{1\}\{\}\int \_\{\mathbb \{R\}^\{3\}\}|\nabla u|^\{2\}+\frac\{1\}\{^\{3\}\}\int \_\{\mathbb \{R\}^\{3\}\}V(x)u^\{2\}\right)\left[-^\{2\}\Delta u + V(x)u \right], \]ℒ ε u = M 1 ε ∫ IR 3 | ∇ u | 2 + 1 ε 3 ∫ IR 3 V ( x ) u 2 [ − ε 2 Δ u + V ( x ) u ] ,M : IR+ → IR+ and V : IR3 → IR are continuous functions which verify some hypotheses.},
author = {Figueiredo, Giovany M., Santos, João R.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {penalization method; Schrödinger–Kirchhoff type problem; Lusternik–Schnirelmann theory; Moser iteration; Schrödinger-Kirchhoff type problem; Lusternik-Schnirelmann theory},
language = {eng},
number = {2},
pages = {389-415},
publisher = {EDP-Sciences},
title = {Multiplicity and concentration behavior of positive solutions for a Schrödinger–Kirchhoff type problem via penalization method},
url = {http://eudml.org/doc/272888},
volume = {20},
year = {2014},
}

TY - JOUR
AU - Figueiredo, Giovany M.
AU - Santos, João R.
TI - Multiplicity and concentration behavior of positive solutions for a Schrödinger–Kirchhoff type problem via penalization method
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2014
PB - EDP-Sciences
VL - 20
IS - 2
SP - 389
EP - 415
AB - In this paper we are concerned with questions of multiplicity and concentration behavior of positive solutions of the elliptic problem\[ (P_{})\hspace*{113.81102pt} \left\lbrace \begin{array}{rcl} \mathcal {L}_{}u=f(u) \ \ \mbox{in} \ \ \mathbb {R}^{3},\\[1.5mm] u&gt;0 \ \ \mbox{in} \ \ \mathbb {R}^{3},\\[1.5mm] u \in H^{1}(\mathbb {R}^3), \end{array} \right. \]( P ε ) ℒ ε u = f ( u ) in IR 3 , u &gt; 0 in IR 3 , u ∈ H 1 ( IR 3 ) , whereε is a small positive parameter, f : ℝ → ℝ is a continuous function,\[ \mathcal {L}_{} \]ℒ ε is a nonlocal operator defined by\[ \mathcal {L}_{}u=M\left(\frac{1}{}\int _{\mathbb {R}^{3}}|\nabla u|^{2}+\frac{1}{^{3}}\int _{\mathbb {R}^{3}}V(x)u^{2}\right)\left[-^{2}\Delta u + V(x)u \right], \]ℒ ε u = M 1 ε ∫ IR 3 | ∇ u | 2 + 1 ε 3 ∫ IR 3 V ( x ) u 2 [ − ε 2 Δ u + V ( x ) u ] ,M : IR+ → IR+ and V : IR3 → IR are continuous functions which verify some hypotheses.
LA - eng
KW - penalization method; Schrödinger–Kirchhoff type problem; Lusternik–Schnirelmann theory; Moser iteration; Schrödinger-Kirchhoff type problem; Lusternik-Schnirelmann theory
UR - http://eudml.org/doc/272888
ER -

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