Ground states for $p$-fractional Choquard-type equations with critical local nonlinearity and doubly critical nonlocality

Abstract

We consider a $p$-fractional Choquard-type equation \[ (-Δ)_p^s u+a|u|^{p-2}u=b(K\ast F(u))F'(u)+\varepsilon_g |u|^{p_g-2}u \quad\text{in ${\mathbb{R}}^N$}, \] where $0<s<1<p<p_g\le p_s^{\ast }$, $N\ge \max \{2ps+\alpha ,p^{2}s\}$, $a,b,{\epsilon }_g\in (0,\infty )$, $K(x)=|x{|}^{-(N-\alpha )}$, $\alpha \in (0,N)$, and $F$ is a doubly critical nonlinearity in the sense of the Hardy-Littlewood-Sobolev inequality. It is noteworthy that the local nonlinearity may also have critical growth. Combining Brezis-Nirenberg's method with some new ideas, we obtain the ground state solutions via the mountain pass lemma and a generalized Lions-type theorem.

14 pages