Multiple nodal solutions of quadratic Choquard equations with perturbation

Abstract

In this paper, we consider the Choquard equations with a local perturbation $-\mathrm{\Delta }u+u=\left({\int }_{{\mathbb{R}}^N}\frac{|u\left(y\right){|}^2}{|x-y{|}^{N-\alpha }}\mathrm{d}y\right)u+|u{|}^{q-2}u\text{in}{\mathbb{R}}^N,$ where $N\ge 3,\alpha \in \left(\right(N-4{)}^{+},N),q\in (2,2N/(N-2))$. By using variational method and approximating approach, we prove that for any given positive integer k, the above equation has a least energy radial solution changing sign exactly k times. This solution is constructed as the limit of such solutions for the following Choquard equations $-\mathrm{\Delta }u+u=\left({\int }_{{\mathbb{R}}^N}\frac{|u\left(y\right){|}^p}{|x-y{|}^{N-\alpha }}\mathrm{d}y\right)|u{|}^{p-2}u+|u{|}^{q-2}u\text{in}{\mathbb{R}}^N,$ as $p\searrow 2_{+}$. Our result improves and extends the previous results in the literature.

COMMUNICATED BY:

• G. Molica Bisci

Keywords:

• Quadratic Choquard equations
• variational methods
• sign-changing solutions

AMS SUBJECT CLASSIFICATIONS:

• 35J20
• 35J57
• 35Q35

Disclosure statement

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

Additional information

Funding

This work is partially supported by Natural Science Foundation of Hunan Province (Grant No. 2018JJ3136), and Scientific Research Fund of Hunan Provincial Education Department (No. 19C0781,18C0293,19A179).