Bulletin of the Malaysian Mathematical Sciences Society ( IF 1.2 ) Pub Date : 2021-05-06 , DOI: 10.1007/s40840-021-01127-6 Liejun Shen
We study the existence and asymptotic behavior of least energy sign-changing solutions for a N-Laplacian equation of Kirchhoff type with critical exponential growth in \(\mathbb {R}^N\)
$$\begin{aligned} \left\{ \begin{aligned}&- \bigg (a+b\int _{\mathbb {R}^N}|\nabla u|^N\mathrm{d}x\bigg )\Delta _N u+V(|x|) |u|^{N-2}u= f(|x|,u), \\&u\in W^{1,N}(\mathbb {R}^N), \\ \end{aligned} \right. \end{aligned}$$where \(a,b>0\) are constants, \(\Delta _Nu=\text {div}(|\nabla u|^{N-2}\nabla u)\), and V(x) is a smooth function. Under some suitable assumptions on \(f\in C(\mathbb {R}^N\times \mathbb {R})\), we apply the constraint minimization argument to establish a least energy sign-changing solution \(u_b\) with precisely two nodal domains. Moreover, we show that the energy of \(u_b\) is strictly larger than two times of the ground state energy and analyze the asymptotic behavior of \(u_b\) as \(b\searrow 0^+\). Our results generalize the existing ones to the N-Kirchhoff equation with critical growth.
中文翻译：
具有临界指数增长的$$ \ mathbb {R} ^ N $$ RN的N-Kirchhoff方程的符号转换解
我们研究具有临界指数增长\（\ mathbb {R} ^ N \）的Kirchhoff型N-Laplacian方程的最小能量符号转换解的存在性和渐近行为
$$ \ begin {aligned} \ left \ {\ begin {aligned}＆-\ bigg（a + b \ int _ {\ mathbb {R} ^ N} | \ nabla u | ^ N \ mathrm {d} x \ bigg）\ Delta _N u + V（| x |）| u | ^ {N-2} u = f（| x |，u），\\＆u \ in W ^ {1，N}（\ mathbb {R } ^ N），\\ \ end {aligned} \ right。\ end {aligned} $$其中\（a，b> 0 \）是常量，\（\ Delta _Nu = \ text {div}（| \ nabla u | ^ {N-2} \ nabla u）\），而V（x）是a功能流畅。在\（f \ in C（\ mathbb {R} ^ N \ times \ mathbb {R}）\）的一些适当假设下，我们应用约束最小化参数来建立最小能量符号变化的解决方案\（u_b \）恰好有两个节点域。此外，我们证明\（u_b \）的能量严格大于基态能量的两倍，并以\（b \ searrow 0 ^ + \）的形式分析\（u_b \）的渐近行为。我们的结果将现有的结果推广到具有临界增长的N-Kirchhoff方程。
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