Singularly perturbed Choquard equations with nonlinearity satisfying Berestycki-Lions assumptions

Abstract

In the present paper, we consider the following singularly perturbed problem: \begin{equation*} \left\{ \begin{array}{ll} -\varepsilon^2\triangle u+V(x)u=\varepsilon^{-\alpha}(I_{\alpha}*F(u))f(u), & x\in \R^N; u\in H^1(\R^N), \end{array} \right. \end{equation*} where $\varepsilon>0$ is a parameter, $N\ge 3$, $\alpha\in (0, N)$, $F(t)=\int_{0}^{t}f(s)\mathrm{d}s$ and $I_{\alpha}: \R^N\rightarrow \R$ is the Riesz potential. By introducing some new tricks, we prove that the above problem admits a semiclassical ground state solution ($\varepsilon\in (0,\varepsilon_0)$) and a ground state solution ($\varepsilon=1$) under the general "Berestycki-Lions assumptions" on the nonlinearity $f$ which are almost necessary, as well as some weak assumptions on the potential $V$. When $\varepsilon=1$, our results generalize and improve the ones in [V. Moroz, J. Van Schaftingen, T. Am. Math. Soc. 367 (2015) 6557-6579] and [H. Berestycki, P.L. Lions, Arch. Rational Mech. Anal. 82 (1983) 313-345] and some other related literature. In particular, our approach is useful for many similar problems.