Existence Results for Fractional Choquard Equations with Critical or Supercritical Growth

Abstract

AbstractIn this paper, we study the following fractional Choquard equation with critical or supercritical growth $$\begin{aligned} \ (-\Delta )^su+V(x)u=f(x,u)+\lambda \left[ |x|^{-\mu }*|u|^p\right] p|u|^{p-2}u, \quad x \in {\mathbb {R}}^N, \end{aligned}$$ ( - Δ ) s u + V ( x ) u = f ( x , u ) + λ | x | - μ ∗ | u | p p | u | p - 2 u , x ∈ R N , where $$0<s<1$$ 0 < s < 1 , $$(-\Delta )^s$$ ( - Δ ) s denotes the fractional Laplacian of order s, $$N>2s$$ N > 2 s , $$0<\mu <2s$$ 0 < μ < 2 s and $$p\ge 2_{\mu ,s}^*:=\frac{2N-\mu }{N-2s}$$ p ≥ 2 μ , s ∗ : = 2 N - μ N - 2 s , which is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. Under some suitable conditions, we prove that the equation admits a nontrivial solution for small $$\lambda >0$$ λ > 0 by variational methods, which extends results in Bhattarai in J. Differ. Equ. 263, 3197–3229 (2017).