Ground state solutions for a class of semilinear elliptic systems with sum of periodic and vanishing potentials
Keywords
Semilinear elliptic systems, ground state, periodic potential, localized potential, variational methodsAbstract
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.References
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