Topological Methods in Nonlinear Analysis

Ground state solutions for a class of semilinear elliptic systems with sum of periodic and vanishing potentials

Guofeng F. Che and Haibo B. Chen

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Abstract

In this paper, we consider the following semilinear elliptic systems: $$ \begin{cases} -\Delta u+V(x)u=F_{u}(x, u, v)-\Gamma(x)|u|^{q-2}u & \mbox{in }\mathbb{R}^{N},\\ -\Delta v+V(x)v=F_{v}(x, u, v)-\Gamma(x)|v|^{q-2}v & \mbox{in }\mathbb{R}^{N},\\ \end{cases} $$ where $q\in[2,2^{*})$, $V=V_{{\rm per}}+V_{{\rm loc}}\in L^{\infty}(\mathbb{R}^{N})$ is the sum of a periodic potential $V_{{\rm per}}$ and a localized potential $V_{{\rm loc}}$ and $\Gamma\in L^{\infty}(\mathbb{R}^{N})$ is periodic and $\Gamma(x)\geq0$ for almost every $x\in\mathbb{R}^{N}$. Under some appropriate assumptions on $F$, we investigate the existence and nonexistence of ground state solutions for the above system. Recent results from the literature are improved and extended.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 51, Number 1 (2018), 215-242.

Dates
First available in Project Euclid: 27 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1511751649

Digital Object Identifier
doi:10.12775/TMNA.2017.046

Mathematical Reviews number (MathSciNet)
MR3784743

Zentralblatt MATH identifier
06887979

Citation

Che, Guofeng F.; Chen, Haibo B. Ground state solutions for a class of semilinear elliptic systems with sum of periodic and vanishing potentials. Topol. Methods Nonlinear Anal. 51 (2018), no. 1, 215--242. doi:10.12775/TMNA.2017.046. https://projecteuclid.org/euclid.tmna/1511751649


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