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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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.

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

First available in Project Euclid: 27 November 2017

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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.

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