Abstract
We are concerned with ground-state solutions for the following Kirchhoff type equation with critical nonlinearity: \t\t\t{−(ε2a+εb∫R3|∇u|2)Δu+V(x)u=λW(x)|u|p−2u+|u|4uin R3,u>0,u∈H1(R3),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\textstyle\\begin{cases} - ({\\varepsilon^{2}}a + \\varepsilon b\\int_{{\\mathbb{R}^{3}}} {{{ \\vert {\\nabla u} \\vert }^{2}}} )\\Delta u + V(x)u = \\lambda W(x){ \\vert u \\vert ^{p - 2}}u + { \\vert u \\vert ^{4}}u\\quad {\\text{in }}{\\mathbb{R}^{3}} ,\\\\ u > 0, \\quad\\quad u \\in{H^{1}}({\\mathbb{R}^{3}}) , \\end{cases} $$\\end{document} where ε is a small positive parameter, a,b>0, lambda > 0, 2 < p le4, V and W are two potentials. Under proper assumptions, we prove that, for varepsilon > 0 sufficiently small, the above problem has a positive ground-state solution {u_{varepsilon}} by using a monotonicity trick and a new version of global compactness lemma. Moreover, we use another global compactness method due to Gui (Commun. Partial Differ. Equ. 21:787-820, 1996) to show that {u_{varepsilon}} is concentrated around a set which is related to the set where the potential V(x) attains its global minima or the set where the potential W(x) attains its global maxima as varepsilon to0.
Highlights
1 Introduction In this paper, we study the following Kirchhoff type equation with critical nonlinearity:
After the pioneer work of Lions [ ], where a functional analysis approach was proposed, the Kirchhoff type equations began to arouse the attention of researchers
The main difficulties in proving Theorem . lie in the fact that the nonlinearity λW (x)|u|p– u + |u| u ( < p ≤ ) does not satisfy the (AR) condition, which prevents us from obtaining a bounded (PS) sequence and the lack of compactness due to the unboundedness of the domain R and the nonlinearity with the critical Sobolev growth
Summary
We study the following Kirchhoff type equation with critical nonlinearity:. In [ ], Ding and Liu studied the existence and concentration of semiclassical solutions for Schrödinger equations with magnetic fields under the condition (P ). Lie in the fact that the nonlinearity λW (x)|u|p– u + |u| u ( < p ≤ ) does not satisfy the (AR) condition, which prevents us from obtaining a bounded (PS) sequence and the lack of compactness due to the unboundedness of the domain R and the nonlinearity with the critical Sobolev growth. For ε > small but fixed, for almost every μ ∈ [ – δ , ], there exists a bounded (PS)cε,μ sequence {un} for Iε,μ. Step : Fix ε > small, choose a sequence {μn} ⊂ [ – δ , ] satisfying μn → , we get a sequence of nontrivial critical points {uε,μn } of Iε,μn with Iε,μn (uε,μn ) = cε,μn. Similar to the argument in Step , we see that ∃wε ∈ H (R ) such that un → wε in H R
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