ABSTRACT

We consider the following Kirchhoff type equation: $-\left(a+b{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }u{|}^2\mathrm{d}x\right)\mathrm{\Delta }u=g\left(u\right),\mathrm{i}\mathrm{n}{\mathbb{R}}^4,$ where a, b are positive constants and $g\in C(\mathbb{R},\mathbb{R})$. Under the critical growth assumptions on g, we obtain the existence of least energy solutions by studying the associated minimization problem with a new constraint. The mountain pass characterization of least energy solutions is also investigated without the monotonicity assumption on $g\left(s\right)/s$. Our results present some new observations.

中文翻译：

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关于ℝ4中涉及临界增长的基尔霍夫型方程的评论

摘要

我们考虑以下基尔霍夫型方程：$-\left(\mathrm{一种}+b{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }你{|}^2\mathrm{d}X\right)\mathrm{\Delta }你=G\left(你\right),\mathrm{一世}\mathrm{n}{\mathbb{R}}^4,$其中a , b是正常数和$G\in C(\mathbb{R},\mathbb{R})$. 在g的临界增长假设下，我们通过研究具有新约束的相关最小化问题来获得最小能量解的存在。在没有单调性假设的情况下，还研究了最小能量解的山口表征$G\left(s\right)/s$. 我们的结果提出了一些新的观察结果。