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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X1630236X&_mathId=si2.gif&_user=111111111&_pii=S0362546X1630236X&_rdoc=1&_issn=0362546X&md5=07e4e72d63b7a83231f7d78b8736cb0b" title="Click to view the MathML source">a≥0class="mathContainer hidden">class="mathCode">$a\ge 0$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X1630236X&_mathId=si3.gif&_user=111111111&_pii=S0362546X1630236X&_rdoc=1&_issn=0362546X&md5=fb59ca7b5681e621d619720d84aeb3a9" title="Click to view the MathML source">p∈(1,n)class="mathContainer hidden">class="mathCode">$p\in (1,n)$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X1630236X&_mathId=si4.gif&_user=111111111&_pii=S0362546X1630236X&_rdoc=1&_issn=0362546X&md5=609a4706022c45810d0bbaf053221230" title="Click to view the MathML source">μ∈Rclass="mathContainer hidden">class="mathCode">$\mu \in \mathbb{R}$. class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X1630236X&_mathId=si5.gif&_user=111111111&_pii=S0362546X1630236X&_rdoc=1&_issn=0362546X&md5=23738ff89bab2cd8480e6cbef02a0e60" title="Click to view the MathML source">V(x)class="mathContainer hidden">class="mathCode">$V\left(x\right)$ is a trapping type potential, e.g., class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X1630236X&_mathId=si6.gif&_user=111111111&_pii=S0362546X1630236X&_rdoc=1&_issn=0362546X&md5=6bbe680d6ac873083aca90ad54851570" title="Click to view the MathML source">inf_x∈RⁿV(x)<lim_|x|→+∞V(x)class="mathContainer hidden">class="mathCode">${inf}_{x\in {\mathbb{R}}^n}V\left(x\right)<{lim}_{\left|x\right|\to +\infty }V\left(x\right)$. By using constrained variational methods, we proved that there is class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X1630236X&_mathId=si7.gif&_user=111111111&_pii=S0362546X1630236X&_rdoc=1&_issn=0362546X&md5=2f52be265a7dcafcc3a3ef5618db0b28" title="Click to view the MathML source">a^∗>0class="mathContainer hidden">class="mathCode">$a^{\ast }>0$, which can be given explicitly, such that problem (P) has a ground state class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X1630236X&_mathId=si8.gif&_user=111111111&_pii=S0362546X1630236X&_rdoc=1&_issn=0362546X&md5=c94c15f0aca45c2e350875fbb552a162" title="Click to view the MathML source">uclass="mathContainer hidden">class="mathCode">$u$ with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X1630236X&_mathId=si9.gif&_user=111111111&_pii=S0362546X1630236X&_rdoc=1&_issn=0362546X&md5=3902ad63af5e7f9665f497235f24a584" title="Click to view the MathML source">|u|_L^p=1class="mathContainer hidden">class="mathCode">${\left|u\right|}_{L^p}=1$ for some class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X1630236X&_mathId=si4.gif&_user=111111111&_pii=S0362546X1630236X&_rdoc=1&_issn=0362546X&md5=609a4706022c45810d0bbaf053221230" title="Click to view the MathML source">μ∈Rclass="mathContainer hidden">class="mathCode">$\mu \in \mathbb{R}$ and all class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X1630236X&_mathId=si11.gif&_user=111111111&_pii=S0362546X1630236X&_rdoc=1&_issn=0362546X&md5=ca1325a710870acaeefd2a32cf03524f" title="Click to view the MathML source">a∈[0,a^∗)class="mathContainer hidden">class="mathCode">$a\in [0,a^{\ast })$, but (P) has no this kind of ground state if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X1630236X&_mathId=si12.gif&_user=111111111&_pii=S0362546X1630236X&_rdoc=1&_issn=0362546X&md5=eb07468b22bae65906d387cf535bd113" title="Click to view the MathML source">a≥a^∗class="mathContainer hidden">class="mathCode">$a\ge a^{\ast }$. 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X1630236X&_mathId=si5.gif&_user=111111111&_pii=S0362546X1630236X&_rdoc=1&_issn=0362546X&md5=23738ff89bab2cd8480e6cbef02a0e60" title="Click to view the MathML source">V(x)class="mathContainer hidden">class="mathCode">$V\left(x\right)$ and blows up if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X1630236X&_mathId=si14.gif&_user=111111111&_pii=S0362546X1630236X&_rdoc=1&_issn=0362546X&md5=fd40656c2c30d21db04cff900c361444" title="Click to view the MathML source">a↗a^∗class="mathContainer hidden">class="mathCode">$a\nearrow a^{\ast }$. The optimal rate of blowup is obtained for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X1630236X&_mathId=si5.gif&_user=111111111&_pii=S0362546X1630236X&_rdoc=1&_issn=0362546X&md5=23738ff89bab2cd8480e6cbef02a0e60" title="Click to view the MathML source">V(x)class="mathContainer hidden">class="mathCode">$V\left(x\right)$ being a polynomial type potential.