ABSTRACT

In this article, we consider the following new Kirchhoff-type problem: $\begin{array}{rl}-\left(a-b{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }u{|}^2\mathrm{d}x\right)\mathrm{\Delta }u& =|u{|}^{p-2},\mathrm{i}\mathrm{n}\mathrm{\Omega },\\ u& =0,\mathrm{o}\mathrm{n}\mathrm{\partial }\mathrm{\Omega },\end{array}$ where a and b are positive constants, $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded domain with $C^1$ boundary $\mathrm{\partial }\mathrm{\Omega }$, $p\in [2,2^{\ast })$ with $2^{\ast }=2N/\left(N-2\right)$ if $N\ge 3$, and $2^{\ast }=+\mathrm{\infty }$ if N = 1, 2. We show that the problem possesses infinitely many sign-changing solutions by using combination of invariant sets of descent flow and the Ljusternik–Schnirelman type minimax method. And an example for p = 2 is illustrated our results.

中文翻译：

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具有亚临界指数的新基尔霍夫型方程的无穷多解

摘要

在本文中，我们考虑以下新的 Kirchhoff 型问题：$\begin{array}{rl}-\left(\mathrm{一种}-b{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }你{|}^2\mathrm{d}X\right)\mathrm{\Delta }你& =|你{|}^{p-2},\mathrm{一世}\mathrm{n}\mathrm{\Omega },\\ 你& =0,\mathrm{○}\mathrm{n}\mathrm{\partial }\mathrm{\Omega },\end{array}$其中a和b是正常数，$\mathrm{\Omega }\subset {\mathbb{R}}^{ñ}$是一个有界域$C^1$边界$\mathrm{\partial }\mathrm{\Omega }$,$p\in [2,2^{*})$和$2^{*}=2ñ/\left(ñ-2\right)$如果$ñ\ge 3$， 和$2^{*}=+\mathrm{\infty }$如果N  = 1, 2。我们通过使用下降流的不变集和 Ljusternik-Schnirelman 型极小极大方法的组合来证明该问题具有无限多个符号变化解。p  = 2 的示例说明了我们的结果。