Existence of solutions for a nonlinear Choquard equation with potential vanishing at infinity

We study the following class of nonlinear Choquard equation, $$ -Δu +V(x)u =\Big( \frac{1}{|x|^μ}\ast F(u)\Big)f(u) \quad \mbox{in} \quad \R^N, $$ where $0<μ<N$, $N \geq 3$, $V$ is a continuous real function and $F$ is the primitive function of $f$. Under some suitable assumptions on the potential $V$, which include the case $V(\infty)=0$, that is, $V(x)\to 0$ as $|x|\to +\infty$, we prove existence of a nontrivial solution for the above equation by penalization method.

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