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AbstractWe use the non-Nehari manifold method to deal with the system {−Δu+V(x)u+ϕu=(∫R3Q(y)F(u(y))|x−y|μdy)Q(x)f(u(x)),x∈R3,−Δϕ=u2,u∈H1(R3),$$ \textstyle\begin{cases} -\Delta u+V(x)u+\phi u= (\int_{\mathbb{R}^{3}}\frac {Q(y)F(u(y))}{|x-y|^{\mu}}\,dy )Q(x)f(u(x)),\quad x\in\mathbb{R}^{3}, \\ -\Delta\phi=u^{2}, \quad u \in H^{1}(\mathbb{R}^{3}), \end{cases} $$ where V(x)$V(x)$… Click to show full abstract
AbstractWe use the non-Nehari manifold method to deal with the system {−Δu+V(x)u+ϕu=(∫R3Q(y)F(u(y))|x−y|μdy)Q(x)f(u(x)),x∈R3,−Δϕ=u2,u∈H1(R3),$$ \textstyle\begin{cases} -\Delta u+V(x)u+\phi u= (\int_{\mathbb{R}^{3}}\frac {Q(y)F(u(y))}{|x-y|^{\mu}}\,dy )Q(x)f(u(x)),\quad x\in\mathbb{R}^{3}, \\ -\Delta\phi=u^{2}, \quad u \in H^{1}(\mathbb{R}^{3}), \end{cases} $$ where V(x)$V(x)$ and Q(x)$Q(x)$ are periodic and asymptotically periodic in x. Under some mild conditions on f, we establish the existence of the Nehari type ground state solutions in two cases: the periodic one and the asymptotically periodic case.
Journal Title: Boundary Value Problems
Year Published: 2018
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