Existence of solutions for fractional Kirchhoff–Schrödinger–Poisson equations via Morse theory

Abstract

This paper is devoted to the following fractional Kirchhoff–Schrödinger–Poisson equations: $\{\begin{array}{l}{\displaystyle (a+b[u{]}_s^2)(-\mathrm{\Delta }{)}^{s}u+V(x)u+\varphi (x)u}\\ =-\lambda |u{|}^{q-2}u+f(x,u)& \mathrm{in}\mathrm{\Omega },\\ {\displaystyle (-\mathrm{\Delta }{)}^t\varphi (x)=u^2}& \mathrm{in}\mathrm{\Omega },\\ u=0& \mathrm{on}\mathrm{\partial }\mathrm{\Omega },\end{array}$ where $(-\mathrm{\Delta }{)}^s$ is the fractional Laplacian, $s,t\in (0,1)$, $2t+4s>3,a>0,b>0,1<q<2$, $V:\mathrm{\Omega }\to {\mathbb{R}}^{+}$ is continuous, Ω is a smooth bounded domain in ${\mathbb{R}}^3$. With the help of Morse theory, we prove that the Dirichlet boundary value problem has at least a weak nontrivial solution.

Keywords:

• Fractional Kirchhoff–Schrödinger–Poisson equations
• Dirichlet boundary
• Morse theory

AMS Subject Classifications:

• 35R11
• 35B38
• 58E05

Disclosure statement

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

Additional information

Funding

This work was supported by National Natural Science Foundation of China (NSFC): (12161038), Science and Technology project of Jiangxi provincial Department of Education (Grant No. GJJ212204).