On a nonhomogeneous Kirchhoff-type elliptic problem with critical exponential in dimension two

ABSTRACT

In this article, the Ekeland variational principle, mountain pass theorem and Trudinger–Moser inequality are applied to establish the existence and multiplicity of solutions for the following nonhomogeneous Kirchhoff-type elliptic problem: $\begin{array}{ll}-m\left({\int }_{\mathrm{\Omega }}|\mathrm{\nabla }u{|}^2\mathrm{d}x\right)\mathrm{\Delta }u=f(x,u)+ϵh\left(x\right)& \text{in}\mathrm{\Omega };\\ u=0& \text{on}\mathrm{\partial }\mathrm{\Omega },\end{array}$ where Ω is a smooth bounded domain in ${\mathbb{R}}^2$, $m:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is a Kirchhoff function, $f:\mathrm{\Omega }\times \mathbb{R}\to \mathbb{R}$ is a continuous function, $h\in {\left(W_0^{1,2}\left(\mathrm{\Omega }\right)\right)}^{\ast }=W^{-1,2}$, $h\ge 0$, $h\not\equiv 0$, and ε is a small positive parameter. The nonlinearity term $f(x,s)$ behaves like $e^{{\alpha }_0{s}^2}$ when $|s|\to \mathrm{\infty }$ for some ${\alpha }_0>0$.

KEYWORDS:

• Nonhomogeneous elliptic problem
• Kirchhoff-type problem
• critical exponential growth
• Trudinger–Moser inequality

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35J20
• 35J60

Disclosure statement

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

Additional information

Funding

The authors were supported by the National Natural Science Foundation of China (11971392), Fundamental Research Funds for the Central Universities, and Natural Science Foundation of Chongqing, China (cstc2019jcyjjqX0022).