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 a≥0$a\ge 0$, p∈(1,n)$p\in (1,n)$ and baf053221230" title="Click to view the MathML source">μ∈R$\mu \in \mathbb{R}$. bab2cd8480e6cbef02a0e60" title="Click to view the MathML source">V(x)$V\left(x\right)$ is a trapping type potential, e.g., inf_x∈RⁿV(x)<lim_|x|→+∞V(x)${inf}_{x\in {\mathbb{R}}^n}V\left(x\right)<{lim}_{\left|x\right|\to +\infty }V\left(x\right)$. By using constrained variational methods, we proved that there is a^&lowast;>0$a^{\&<font\; color="red">lowast</font>;}>0$, which can be given explicitly, such that problem (P) has a ground state u$u$ with |u|_L^p=1${\left|u\right|}_{L^p}=1$ for some baf053221230" title="Click to view the MathML source">μ∈R$\mu \in \mathbb{R}$ and all a∈[0,a^&lowast;)$a\in [0,a^{\&<font\; color="red">lowast</font>;})$, but (P) has no this kind of ground state if bae65906d387cf535bd113" title="Click to view the MathML source">a≥a^&lowast;$a\ge a^{\&<font\; color="red">lowast</font>;}$. 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 bab2cd8480e6cbef02a0e60" title="Click to view the MathML source">V(x)$V\left(x\right)$ and blows up if a↗a^&lowast;$a\nearrow a^{\&<font\; color="red">lowast</font>;}$. The optimal rate of blowup is obtained for bab2cd8480e6cbef02a0e60" title="Click to view the MathML source">V(x)$V\left(x\right)$ being a polynomial type potential.