## Ground state solutions for a class of strongly indefinite Choquard equations.(English)Zbl 1440.35126

Summary: In this paper, we study the existence and concentration of ground state solution for the Choquard equation
$\begin{cases}-\Delta u+V(x)u=\left(\int_{\mathbb{R}^N}\frac{A(\epsilon y)|u(y)|^p}{|x-y|^\mu}\text{d}y\right)A(\epsilon x)|u|^{p-2}u\quad\text{in }\mathbb{R}^N,\\u\in H^1(\mathbb{R}^N),\end{cases}$
where $$N\ge 2$$, $$0<\mu <2$$, $$\epsilon$$ is a positive parameter. $$V$$ is a $$\mathbb{Z}^N$$-periodic function, and 0 lies in a gap of the spectrum of $$-\Delta+V$$. $$A\in C(\mathbb{R}^N)$$ satisfies
\begin{aligned}0<\inf\limits_{x\in\mathbb{R}^N}A(x)\le\lim\limits_{|x|\rightarrow+\infty}A(x) <\sup\limits_{x\in\mathbb{R}^N}A(x).\end{aligned}

### MSC:

 35J61 Semilinear elliptic equations 35J50 Variational methods for elliptic systems 58E30 Variational principles in infinite-dimensional spaces 35A01 Existence problems for PDEs: global existence, local existence, non-existence
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