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

Guofeng Che, Haibo Chen



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 $q\in[2,2^{*})$, $V=V_{\rom{per}}+V_{\rom{loc}}\in L^{\infty}(\mathbb{R}^{N})$ is the sum of a periodic potential $V_{\rom{per}}$ and a localized potential $V_{\rom{loc}}$ 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.


Semilinear elliptic systems; ground state; periodic potential; localized potential; variational methods

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