Positive solutions for Schrödinger system with asymptotically periodic potentials

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• 作者：Jun Wang^a ; ^{wangmath2011@126.com" class="auth_mail" title="E-mail the corresponding author} ; Qing He^a ; ^{18252586526@126.com" class="auth_mail" title="E-mail the corresponding author} ; Lu Xiao^b ; ^{hnlulu@126.com" class="auth_mail" title="E-mail the corresponding author} ; Fubao Zhang^c ; ^{zhangfubao@seu.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Weakly coupled Schrö ; dinger system ; Positive solutions ; Variational methods
• 刊名：Nonlinear Analysis
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：134
• 期：Complete
• 页码：215-235
• 全文大小：686 K

文摘

This paper is concerned with the following class of elliptic equations where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16000195&_mathId=si2.gif&_user=111111111&_pii=S0362546X16000195&_rdoc=1&_issn=0362546X&md5=f92b45869854f01a804cbec5de7d05f9" title="Click to view the MathML source">u,v∈H¹(R^N)class="mathContainer hidden">class="mathCode">$u,v\in H^1\left({\mathbb{R}}^N\right)$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16000195&_mathId=si3.gif&_user=111111111&_pii=S0362546X16000195&_rdoc=1&_issn=0362546X&md5=7f995773a8562380f481495585d0d28a" title="Click to view the MathML source">N≤3class="mathContainer hidden">class="mathCode">$N\le 3$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16000195&_mathId=si4.gif&_user=111111111&_pii=S0362546X16000195&_rdoc=1&_issn=0362546X&md5=7cfb7c707e7752de8a621d5858fb3c7b" title="Click to view the MathML source">μ,ν,β>0class="mathContainer hidden">class="mathCode">$\mu ,\nu ,\beta >0$ are coupling constants, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16000195&_mathId=si5.gif&_user=111111111&_pii=S0362546X16000195&_rdoc=1&_issn=0362546X&md5=647fe3b2930e134443738aff34a39aa3" title="Click to view the MathML source">λ(x)class="mathContainer hidden">class="mathCode">$\lambda \left(x\right)$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16000195&_mathId=si6.gif&_user=111111111&_pii=S0362546X16000195&_rdoc=1&_issn=0362546X&md5=1a995e35d4177340e7b62181fe138e88" title="Click to view the MathML source">κ(x)class="mathContainer hidden">class="mathCode">$\kappa \left(x\right)$ are asymptotically periodic functions, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16000195&_mathId=si7.gif&_user=111111111&_pii=S0362546X16000195&_rdoc=1&_issn=0362546X&md5=3b4ae6c1b581ad4819cc614d37d7f0ae" title="Click to view the MathML source">fclass="mathContainer hidden">class="mathCode">$f$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16000195&_mathId=si8.gif&_user=111111111&_pii=S0362546X16000195&_rdoc=1&_issn=0362546X&md5=c77161ae1f3030807020db8d2a58e9b9" title="Click to view the MathML source">gclass="mathContainer hidden">class="mathCode">$g$ are continuous functions with subcritical growth. This type of system arises, in particular, in models in Bose–Einstein condensates theory. We prove the existence of positive solution for this weakly coupled system with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16000195&_mathId=si9.gif&_user=111111111&_pii=S0362546X16000195&_rdoc=1&_issn=0362546X&md5=0a0cd0471397a20534e8f878175505e4" title="Click to view the MathML source">β>0class="mathContainer hidden">class="mathCode">$\beta >0$ sufficiently large. Furthermore, we obtain some sufficient conditions for the nonexistence of positive solutions.