Concentrating ground-state solutions for a class of Schödinger-Poisson equations in $\mathbb{R}^3$ involving critical Sobolev exponents

Yi  He; Lu  Lu; Wei Shuai

doi:10.3934/cpaa.2016.15.103

Article Contents

2016, Volume 15, Issue 1: 103-125. Doi: 10.3934/cpaa.2016.15.103

This issue Previous Article Multiple nontrivial solutions to a $p$-Kirchhoff equation Next Article Riemann problem for the relativistic generalized Chaplygin Euler equations

Concentrating ground-state solutions for a class of Schödinger-Poisson equations in $\mathbb{R}^3$ involving critical Sobolev exponents

Yi He^1,,
Lu Lu^2, and
Wei Shuai^3,

1.
School of Mathematics and Statistics, South-Central University For Nationalities, Wuhan, 430074
2.
School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073
3.
Department of Mathematics, Huazhong Normal University, Wuhan, 430079

Received: March 31, 2015

Revised: July 31, 2015

Published: December 2015

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Abstract

We are concerned with standing waves for the following Schrödinger-Poisson equation with critical nonlinearity: \begin{eqnarray} && - {\varepsilon ^2}\Delta u + V(x)u + \psi (x)u = \lambda W(x){\left| u \right|^{p - 2}}u + {\left| u \right|^4}u\;\;{\text{ in }}\mathbb{R}^3, \\ && - {\varepsilon ^2}\Delta \psi = {u^2}\;\;{\text{ in }}\mathbb{R}^3, u>0, u \in {H^1}(\mathbb{R}^3), \end{eqnarray} where $\varepsilon $ is a small positive parameter, $\lambda > 0$, $3 < p \le 4$, $V$ 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 [C. Gui, Commun. Partial Differential Equations 21 (1996) 787-820] to show that ${u_\varepsilon }$ concentrates 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 $\varepsilon \to 0$. As far as we know, the existence and concentration behavior of the positive solutions to the Schrödinger-Poisson equation with critical nonlinearity $g(u): = \lambda W(x)|u{|^{p - 2}}u + |u{|^4}u$ $(3

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Mathematics Subject Classification: Primary: 35J20, 35J60, 35J92.

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