Existence of least energy nodal solutions for a double-phase problem with nonlocal nonlinearity

ABSTRACT

In this paper, we study the following double-phase problem with nonlocal nonlinearity $\begin{array}{r}\{\begin{array}{ll}-\mathrm{div}(|\mathrm{\nabla }u{|}^{p-2}\mathrm{\nabla }u+a(x)|\mathrm{\nabla }u{|}^{q-2}\mathrm{\nabla }u)=(I_{\mu }\ast |u{|}^r)|u{|}^{r-2}u& \mathrm{in}\mathrm{\Omega },\\ u=0& \mathrm{on}\mathrm{\partial }\mathrm{\Omega },\end{array}\end{array}$ where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded domain, $N\ge 2$, 1<p<q<N, $a(x)\ge 0$, $I_{\mu }:{\mathbb{R}}^N\to \mathbb{R}$ is the Riesz potential of order $\mu \in (0,N)$. By using the constrained variational method and Brouwer degree theory, we prove the existence of least energy nodal solutions.

KEYWORDS:

• Double-phase problem
• least energy nodal solutions
• Musielak-Orlicz space
• variational methods

2020 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35J92
• 35J25
• 35J20
• 35J62

Acknowledgments

The authors are grateful to the referee for his/her constructive and valuable comments and suggestions.

Disclosure statement

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

Additional information

Funding

This work was supported by NSFC [grant number 11971095], NSFJL [grant number 20190201136JC].