Ground states for p-fractional Choquard-type equations with doubly or triply critical nonlinearity

Abstract

We consider a p-fractional Choquard-type equation

$$\begin{aligned} (-\varDelta )_p^s u+a|u|^{p-2}u=b(K*F(u))F'(u)+\varepsilon _g |u|^{p_g-2}u \quad \text {in } \mathbb {R}^N, \end{aligned}$$

where \(0<s<1<p<p_g\le p_s^*\), \(N\ge \max \{2ps+\alpha , p^2 s\}\), \(a,b,\varepsilon _g\in (0,\infty )\), \(K(x)= |x|^{-(N-\alpha )}\), \(\alpha \in (0,N)\) and F(u) 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 ground state solutions via the mountain pass lemma and a new generalized Lions-type theorem.