论文标题
具有Trudinger-Moser生长的Kirchhoff类型椭圆系统的解决方案的存在和多样性
Existence and multiplicity of solutions to a Kirchhoff type elliptic system with Trudinger-Moser growth
论文作者
论文摘要
本文介绍了一类涉及Trudinger-Moser指数增长非线性的Kirchhoff类型椭圆系统的解决方案的存在和多样性。我们首先研究了以下系统的解决方案\ begin {eqnarray*} \ left \ {\ arraycolsep = 1.5pt \ begin {array} {ll} - \ big（a_1+b_1 \ | u \ |^{2（θ_1-1）} \ big）ΔU=λH_U（x，x，u，v）\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ω， - \ big（a_2+b_2 \ | v \ |^{2（θ_2-1）} \ big）ΔV=λH_v（x，x，u，u，v）\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ω， u = 0，v = 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \partialΩ， \ end {array} \正确的。 \ end {eqnarray*}其中$ω$是$ \ mathbb {r}^2 $中的一个有界域，带有平稳的边界，\ $ \ | U \ | = \ big（\int_Ω| \ nabla u |^2dx u |^2dx \ big）当$ | s | \ rightarrow \ infty $ for某些$β> 0 $，$ a_1，\ a_2> 0 $，$ b_1，\ b_2> 0 $，$θ_1，\θ_2> 1 $和$λ$是一个正参数。在本文的后面部分，我们还讨论了上述系统的新多重性结果，其正参数由非局部依赖性引起。必须通过某些新技术克服Kirchhoff项以及由于Trudinger-Moser嵌入而导致的相关能量功能的紧凑性。
This paper deals with the existence and multiplicity of solutions for a class of Kirchhoff type elliptic system involving the Trudinger-Moser exponential growth nonlinearities. We first study the existence of solutions for the following system \begin{eqnarray*} \left\{ \arraycolsep=1.5pt \begin{array}{ll} -\big(a_1+b_1\|u\|^{2(θ_1-1)}\big)Δu= λH_u(x,u,v)\ \ \ &\ \mbox{in}\ \ \ Ω,\\[2mm] -\big(a_2+b_2\|v\|^{2(θ_2-1)}\big)Δv= λH_v(x,u,v)\ \ \ &\ \mbox{in}\ \ \ Ω,\\[2mm] u=0, v=0\ \ \ \ &\ \mbox{on}\ \ \ \partialΩ, \end{array} \right. \end{eqnarray*} where $Ω$ is a bounded domain in $\mathbb{R}^2$ with smooth boundary,\ $\|u\|=\big(\int_Ω|\nabla u|^2dx\big)^{1/2}$, $H_u$ and $H_v$ behave like $e^{β|s|^2}$ when $|s|\rightarrow \infty$ for some $β>0$, $a_1,\ a_2>0$, $b_1,\ b_2> 0$, $θ_1,\ θ_2> 1$ and $λ$ is a positive parameter. In the later part of the paper, we also discuss a new multiplicity result for the above system with a positive parameter induced by the nonlocal dependence. The Kirchhoff term and the lack of compactness of the associated energy functional due to the Trudinger-Moser embedding have to be overcome via some new techniques.
