Polyharmonic Kirchhoff-type problem without Ambrosetti–Rabinowitz condition

Abstract

In this paper, we will study the polyharmonic Kirchhoff-type problem with singular exponential nonlinearity without the Ambrosetti–Rabinowitz condition: $\begin{array}{r}\{\begin{array}{ll}{\displaystyle -M\left({\int }_{\mathrm{\Omega }}|{\mathrm{\nabla }}^{m}u{|}^{\frac{n}{m}}\right){\mathrm{\Delta }}_{\frac{n}{m}}^{m}u=\frac{f(x,u)}{|x{|}^{\sigma }}}& \text{in}\mathrm{\Omega },\\ {\displaystyle u=\mathrm{\nabla }u=\cdots ={\mathrm{\nabla }}^{m-1}u=0}& \text{on}\mathrm{\partial }\mathrm{\Omega },\end{array}\end{array}$ where $\mathrm{\Omega }\subset {\mathbb{R}}^n$ is a bounded domain with smooth boundary, $0<\sigma <n,n\ge 2m\ge 2$, M is a Kirchhoff function and $f(x,u)$ has critical exponential growth. We use a suitable version of the Mountain Pass Theorem to prove the existence of a positive ground state solution for this problem.

Keywords:

• Polyharmonic Kirchhoff problem
• critical exponential growth
• singular exponential nonlinearity
• without the (AR) condition

AMS Subject Classifications:

• 35J30
• 35J62
• 35J91

Disclosure statement

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

Additional information

Funding

The research has been supported by National Nature Science Foundation of China (No. 11971392), Natural Science Foundation of Chongqing, China (No. cstc2019jcyjjqX0022) and Fundamental Research Funds for the Central Universities (No. XDJK2019TY001).