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 ${\alpha }_0>0$.

中文翻译：

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关于二维临界指数的非齐次基尔霍夫型椭圆问题

摘要

在本文中，应用 Ekeland 变分原理、山口定理和 Trudinger-Moser 不等式来确定以下非齐次基尔霍夫型椭圆问题的解的存在性和多重性：$\begin{array}{ll}-米\left({\int }_{\mathrm{\Omega }}|\mathrm{\nabla }你{|}^2\mathrm{d}X\right)\mathrm{\Delta }你=F(X,你)+\epsilon H\left(X\right)& \text{在}\mathrm{\Omega };\\ 你=0& \text{在}\mathrm{\partial }\mathrm{\Omega },\end{array}$其中 Ω 是平滑有界域${\mathbb{R}}^2$,$米：{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$是基尔霍夫函数，$F：\mathrm{\Omega }\times \mathbb{R}\to \mathbb{R}$是一个连续函数，$H\in {\left(W_0^{1,2}\left(\mathrm{\Omega }\right)\right)}^{*}=W^{-1,2}$,$H\ge 0$,$H\not\equiv 0$, ε是一个小的正参数。非线性项$F(X,s)$表现得像$e^{{\alpha }_0{s}^2}$什么时候$|s|\to \mathrm{\infty }$对于一些${\alpha }_0>0$.