Singularly perturbed Choquard equations with nonlinearity satisfying Berestycki-Lions assumptions

Abstract

In the present paper, we consider the following singularly perturbed problem:-{-ϵ2Δu+V(x)u = ϵ-α(Iα ∗ F(u))f(u), x ∈ ℝN;u ∈ H1(ℝN), where ϵ > 0 is a parameter, N ≥ 3, α ∈ (0, N), F(t) = ∫0tf(s)ds and Iα ℝN → ℝ is the Riesz potential. By introducing some new tricks, we prove that the above problem admits a semiclassical ground state solution (ϵ ∈ (0, ϵ0)) 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.