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Singularly perturbed Choquard equations with nonlinearity satisfying Berestycki-Lions assumptions. (English) Zbl 1421.35068

Summary: In the present paper, we consider the following singularly perturbed problem: \[\begin{cases} -\varepsilon^2\Delta u+V(x)u=\varepsilon^{-\alpha}(I_{\alpha}*F(u))f(u),\quad x\in \mathbb{R}^N, \\ u\in H^1(\mathbb{R}^N), \end{cases} \] where \(\varepsilon > 0\) is a parameter, \(N \geq 3\), \(\alpha \in (0, N)\), \(F(t) = \int_{0}^{t} f(s)\mathrm{d}(s)\) and \(I_{\alpha}: \mathbb{R}^{N} \rightarrow \mathbb{R}\) 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 and J. van Schaftingen, Trans. Am. Math. Soc. 367, No. 9, 6557–6579 (2015; Zbl 1325.35052)] and [H. Berestycki and P.-L. Lions, Arch. Ration. Mech. Anal. 82, 313–345 (1983; Zbl 0533.35029)] and some other related literature. In particular, we propose a new approach to recover the compactness for a (PS)-sequence, and our approach is useful for many similar problems.

MSC:

35J10 Schrödinger operator, Schrödinger equation
35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35Q55 NLS equations (nonlinear Schrödinger equations)
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