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Existence of solutions for singularly perturbed Schrödinger equations with nonlocal part

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  • In the present paper we study the existence of solutions for a nonlocal Schrödinger equation \begin{eqnarray*} -\varepsilon^2\Delta u +V(x)u =(\int_{R^3} \frac{|u|^{p}}{|x-y|^{\mu}}dy)|u|^{p-2}u, \end{eqnarray*} where $0 < \mu < 3$ and $\frac{6-\mu}{3} < p < {6-\mu}$. Under suitable assumptions on the potential $V(x)$, if the parameter $\varepsilon$ is small enough, we prove the existence of solutions by using Mountain-Pass Theorem.
    Mathematics Subject Classification: Primary: 35J50, 35J60, 35J25.

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