Ground states for $p$-fractional Choquard-type equations with critical local nonlinearity and doubly critical nonlocality (2305.00897v3)

Abstract: We consider a $p$-fractional Choquard-type equation [ (-\Delta)_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 $0<s<1<p<p_g\leq p_s^*$, $N \geq \max{2ps+\alpha,p^2 s}$, $a,b,\varepsilon_g\in (0,\infty)$, $K(x)= |x|^{-(N-\alpha)}$, $\alpha\in (0,N)$, 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.