Semiclassical states to the nonlinear Choquard equation with critical growth

Abstract

We are concerned with the following Choquard equation:

$$ - {\varepsilon ^2}\Delta u + V(x)u = {\varepsilon ^{- \alpha}}({I_\alpha} * F(u)){F^\prime}(u),\,\,\,\,\,\,x \in {\mathbb{R}^N},$$

where N ⩾ 4, α ∈ (0, N), I_α is the Riesz potential and ε > 0 is a small parameter. Note that $F(u): = {1 \over q}|u{|^q} + {1 \over {2_\alpha ^ *}}|u{|^{2_\alpha ^ *}}$, where 2 ^#_α < q < 2 ^*_α , and $2_\alpha ^\sharp : = {{N + \alpha} \over N}$ and $2_\alpha ^ * : = {{N + \alpha} \over {N - 2}}$ are lower and upper critical exponents in the sense of the Hardy–Littlewood–Sobolev inequality. In this paper, we construct a bound-state concentrating at an isolated component of the positive local minimum points of V as ε → 0 for each q ∈ (2 ^#_α , 2 ^*_α ). This result extends some results established in Cingolani–Tanaka [Rev. Mat. Iberoam. 35 (2019)] for the case q < 2 which was seen as an open problem in Moroz–Van Schaftingen [Calc. Var. Partial Differential Equations 52 (2015)]. The proof of the current paper uses variational methods, a truncation technique and a new regularity result that we develop in this work.