The critical Choquard equations with a Kirchhoff type perturbation in bounded domains

Abstract

This paper deals with the following critical Choquard equation with a Kirchhoff type perturbation in bounded domains, $\{\begin{array}{ll}-(1+b\Vert u{\Vert }^2)\mathrm{\Delta u}=\left({\int }_{\mathrm{\Omega }}\frac{u^2(y)}{|x-y{|}^4}\mathrm{d}y\right)u+\mathrm{\lambda u}& \mathrm{in}\mathrm{\Omega },\\ u=0& \mathrm{on}\mathrm{\partial \Omega },\end{array}$where $\mathrm{\Omega }\subset {\mathbb{R}}^N(N\ge 5)$ is a smooth bounded domain and $\Vert \cdot \Vert$ is the standard norm of $H_0^1(\mathrm{\Omega })$. Under the suitable assumptions on the constant $b\ge 0$, we prove the existence of solutions for $0<\lambda \le {\lambda }_1$, where ${\lambda }_1>0$ is the first eigenvalue of $-\mathrm{\Delta }$ on Ω. Moreover, we prove the multiplicity of solutions for $\lambda >{\lambda }_1$ and b>0 in suitable intervals.

Keywords:

• Choquard equation
• Kirchhoff problem
• critical exponent
• Nehari manifold
• existence

Mathematic Subject classifications:

• 35A15
• 35J60

Disclosure statement

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

Additional information

Funding

This research is supported by Key Scientific Research Projects of Colleges and Universities in Henan Province [grant number 23A110018].