Eigenvalue problem for a p-Laplacian equation with trapping potentials

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• 作者：Long-Jiang Gu^a ; Xiaoyu Zeng^b ; Huan-Song Zhou^b ; ^{hszhou2002@sina.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35J60 ; 35J92 ; 35J20 ; 35P30
• 刊名：Nonlinear Analysis
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：148
• 期：Complete
• 页码：212-227
• 全文大小：709 K

文摘

Consider the following eigenvalue problem of p-Laplacian equation where ource">a≥0$a\ge 0$, ource">p∈(1,n)$p\in (1,n)$ and ource">μ∈R. ource">V(x)$V\left(x\right)$ is a trapping type potential, e.g., ource">inf_x∈RⁿV(x)<lim_|x|→+∞V(x). By using constrained variational methods, we proved that there is ource">a^∗>0$a^{\ast }>0$, which can be given explicitly, such that problem (P) has a ground state ource">u$u$ with ource">|u|_L^p=1${\left|u\right|}_{L^p}=1$ for some ource">μ∈R and all ource">a∈[0,a^∗)$a\in [0,a^{\ast })$, but (P) has no this kind of ground state if ource">a≥a^∗$a\ge a^{\ast }$. Furthermore, by establishing some delicate energy estimates we show that the global maximum point of the ground state of problem (P) approaches one of the global minima of ource">V(x)$V\left(x\right)$ and blows up if ource">a↗a^∗$a\nearrow a^{\ast }$. The optimal rate of blowup is obtained for ource">V(x)$V\left(x\right)$ being a polynomial type potential.