论文标题
Musielak-Sobolev空间中具有双重关键的Kirchhoff类型椭圆方程
Kirchhoff type elliptic equations with double criticality in Musielak-Sobolev spaces
论文作者
论文摘要
本文旨在为非本地问题建立一个弱解决方案的存在：\ begin {equination*} \ left \ {\ begin {array} {ll} {ll} -a \ left（\int_Ω\ mathcal {h} ＆= f（x，u）\ \ \ hbox {in} \ \ω，\ \ \\\\ \ hspace {3.3cm} u＆= 0 \ \ \ \ \ \ hbox {on} \ \ \ \ \ \ \ \ \ \ \ partialω，\ end eend {aray {array {array} \ right right。 \ end {equation*}其中$ω\ subseteq \ mathbb {r}^{n}，\，\，n \ geq 2 $是一个有界且平稳的域，包含两个打开和连接的子集$ω_p$和$ω____________________- $δ_ {\ Mathcal {h}} u = \ hbox {div}（h（x，| \ nabla u |）\ nabla u）$是$ \ mathcal {h} $ - laplace操作员。我们假设$δ_ {\ Mathcal {h}} $将$δ_{p（x）} $减少到$ω__{p} $中，到$δ_{n} $ in $ω__{n}中的$δ_{n} $ $ | t |^{p^{\ ast}（x）-2} t $ on $ω__{p} $，AS $ e^{α|为了确定我们的存在结果，在Musielak-Sobolev空间中，我们使用了基于山间通过定理的各种技术。
This paper aims to establish the existence of a weak solution for the non-local problem: \begin{equation*} \left\{\begin{array}{ll} -a\left(\int_Ω\mathcal{H}(x,|\nabla u|)dx \right) Δ_{\mathcal{H}}u &=f(x,u) \ \ \hbox{in} \ \ Ω, \ \ \ \\ \hspace{3.3cm} u &= 0 \ \ \hbox{on} \ \ \partial Ω, \end{array}\right. \end{equation*} where $Ω\subseteq \mathbb{R}^{N},\, N\geq 2$ is a bounded and smooth domain containing two open and connected subsets $Ω_p$ and $Ω_N$ such that $ \barΩ_{p}\cap\barΩ_{N}=\emptyset$ and $Δ_{\mathcal{H}}u=\hbox{div}( h(x,|\nabla u|)\nabla u)$ is the $\mathcal{H}$-Laplace operator. We assume that $Δ_{\mathcal{H}}$ reduces to $ Δ_{p(x)}$ in $Ω_{p}$ and to $ Δ_{N}$ in $Ω_{N},$ the non-linear function $f:Ω\times\mathbb{R}\rightarrow \mathbb{R}$ act as $|t|^{p^{\ast}(x)-2}t$ on $Ω_{p}$ and as $e^{α|t|^{N/(N-1)}}$ on $Ω_{N}$ for sufficiently large $|t|$. To establish our existence results in a Musielak-Sobolev space, we use a variational technique based on the mountain pass theorem.
