Infinitely many sign-changing solutions for Choquard equation with doubly critical exponents

Abstract

In this paper, we consider the following Choquard equation: $\begin{array}{r}-\mathrm{\Delta }u+u=(I_{\alpha }\ast F(u\left)\right)F^{\prime }\left(u\right)\mathrm{i}\mathrm{n}{\mathbb{R}}^N\end{array}$ where $N⩾3$, $\alpha \in (0,N)$, $I_{\alpha }$ is the Riesz potential, and $F\left(u\right):=\frac{1}{p}|u{|}^p+\frac{1}{q}|u{|}^q$, where $p=\frac{N+\alpha }{N}$ and $q=\frac{N+\alpha }{N-2}$ are lower and upper critical exponents in sense of the Hardy–Littlewood–Sobolev inequality. Based on perturbation method and the invariant sets of descending flow, we prove that the above equation possesses infinitely many sign-changing solutions. Our results extend the results in Seok [Nonlinear Choquard equations: doubly critical case. Appl Math Lett. 2018;76:148–156] and Su [New result for nonlinear Choquard equations: doubly critical case. Appl Math Lett. 2020;102(106092):0–7].

Keywords:

• Choquard equation
• sign-changing solutions
• perturbation approach
• invariant sets of descending flow
• critical exponents

AMS SUBJECT CLASSIFICATIONS:

• 35J10
• 35J20
• 35J61

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was partially supported by National Natural Science Foundation of China [grant number 11671403] and the Fundamental Research Funds for the Central Universities of Central South University [grant number 2019zzts210].