Singularly perturbed Choquard equations with nonlinearity satisfying Berestycki-Lions assumptions Article Swipe
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· 2019
· Open Access
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· DOI: https://doi.org/10.1515/anona-2020-0007
· OA: W2964006947
In the present paper, we consider the following singularly perturbed problem: $$\begin{array}{} \displaystyle \left\{ \begin{array}{ll} -\varepsilon^2\triangle u+V(x)u=\varepsilon^{-\alpha}(I_{\alpha}*F(u))f(u), & x\in \mathbb{R}^N; \\ u\in H^1(\mathbb{R}^N), \end{array} \right. \end{array}$$ where ε > 0 is a parameter, N ≥ 3, α ∈ (0, N ), F ( t ) = $\begin{array}{} \int_{0}^{t} \end{array}$ f ( s )d s and I α : ℝ N → ℝ is the Riesz potential. By introducing some new tricks, we prove that the above problem admits a semiclassical ground state solution ( ε ∈ (0, ε 0 )) and a ground state solution ( ε = 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 ε = 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, we propose a new approach to recover the compactness for a (PS)-sequence, and our approach is useful for many similar problems.
