论文标题
在非均匀的Kirchhoff型椭圆系统上，具有单数Trudinger-Moser的生长
On a nonhomogeneous Kirchhoff type elliptic system with the singular Trudinger-Moser growth
论文作者
论文摘要
本文的目的是研究以下Kirchhoff类型椭圆系统的解决方案的多样性 \ begin {eqnarray*} \ left \ {\ arraycolsep = 1.5pt \ begin {array} {ll} -m \ left（\ sum^k_ {j = 1} \ | U_J \ |^2 \ right）ΔU_i= \ frac {f_i（x，u_1，\ ldots，u_k）} {| x | x | x | x |^β}+\ varepsilon h_i h_i h_i（x） i = 1，\ ldots，k，\\ [2mm] u_1 = u_2 = \ cdots = u_k = 0，\ \＆\ mbox {on} \ \ \ \partialΩ， \ end {array} \正确的。 \end{eqnarray*} where $Ω$ is a bounded domain in $\mathbb{R}^2$ containing the origin with smooth boundary, $β\in [0,2)$, $m$ is a Kirchhoff type function, $\|u_j\|^2=\int_Ω|\nabla u_j|^2dx$, $f_i$ behaves like $ e^{βS^2} $当$ | s | \ rightArrow \ for Some $β> 0 $，并且有$ c^1 $函数$ f：ω\ times \ times \ times \ times \ times \ mathbb {r}^k \ to \ mathbb {r Mathbb {r} $ u_1}，\ ldots，\ frac {\ partial f} {\ partial u_k} \ right）= \ left（f_1，\ ldots，f_k \ right）$，$ h_i \ in \ in \ left（\ left）当$ \ varepsilon> 0 $很小时，我们通过使用具有合适的单数Trudinger-Moser不等式的变异方法来为上述系统的多样性建立足够的条件。
The aim of this paper is to study the multiplicity of solutions for the following Kirchhoff type elliptic systems \begin{eqnarray*} \left\{ \arraycolsep=1.5pt \begin{array}{ll} -m\left(\sum^k_{j=1}\|u_j\|^2\right)Δu_i=\frac{f_i(x,u_1,\ldots,u_k)}{|x|^β}+\varepsilon h_i(x),\ \ & \mbox{in}\ \ Ω, \ \ i=1,\ldots,k ,\\[2mm] u_1=u_2=\cdots=u_k=0,\ \ & \mbox{on}\ \ \partialΩ, \end{array} \right. \end{eqnarray*} where $Ω$ is a bounded domain in $\mathbb{R}^2$ containing the origin with smooth boundary, $β\in [0,2)$, $m$ is a Kirchhoff type function, $\|u_j\|^2=\int_Ω|\nabla u_j|^2dx$, $f_i$ behaves like $e^{βs^2}$ when $|s|\rightarrow \infty$ for some $β>0$, and there is $C^1$ function $F: Ω\times\mathbb{R}^k\to \mathbb{R}$ such that $\left(\frac{\partial F}{\partial u_1},\ldots,\frac{\partial F}{\partial u_k}\right)=\left(f_1,\ldots,f_k\right)$, $h_i\in \left(\big(H^1_0(Ω)\big)^*,\|\cdot\|_*\right)$. We establish sufficient conditions for the multiplicity of solutions of the above system by using variational methods with a suitable singular Trudinger-Moser inequality when $\varepsilon>0$ is small.
