Nonlinear Choquard equations: doubly critical case

TL;DR

This paper proves the existence of nontrivial solutions for a class of nonlinear Choquard equations with doubly critical nonlinearities in higher dimensions, expanding understanding of critical phenomena in nonlocal PDEs.

Contribution

It establishes the existence of solutions for doubly critical Choquard equations in dimensions N ≥ 5 under specific conditions, a novel result in the study of such equations.

Findings

Nontrivial solutions exist for N ≥ 5 when α + 4 < N.

Doubly critical nonlinearities are handled in the Choquard framework.

The results extend the theory of critical nonlinear PDEs with nonlocal terms.

Abstract

Consider nonlinear Choquard equations \begin{equation*} \left\{\begin{array}{rcl} -\Delta u +u & = &(I_\alpha*F(u))F'(u) \quad \text{in } \mathbb{R}^N, \\ \lim_{x \to \infty}u(x) & = &0, \end{array}\right. \end{equation*} where $I_\alpha$ denotes Riesz potential and $\alpha \in (0, N)$. In this paper, we show that when $F$ is doubly critical, i.e. $F(u) = \frac{N}{N+\alpha}|u|^{\frac{N+\alpha}{N}}+\frac{N-2}{N+\alpha}|u|^{\frac{N+\alpha}{N-2}}$, the nonlinear Choquard equation admits a nontrivial solution if $N \geq 5$ and $\alpha + 4 < N$.