Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in

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• 作者：Gongbao Li ; ^{ligb@mail.ccnu.edu.cn" class="auth_mail} ; Hongyu Ye
• 关键词：Kirchhoff equation ; Ground state solutions ; Pohoz虇aev type identity ; Variational methods
• 刊名：Journal of Differential Equations
• 出版年：15 July 2014
• 年：2014
• 卷：257
• 期：2
• 页码：566-600
• 全文大小：410 K

文摘

In this paper, we study the following nonlinear problem of Kirchhoff type with pure power nonlinearities: where mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039614001557&_mathId=si3.gif&_user=111111111&_pii=S0022039614001557&_rdoc=1&_issn=00220396&md5=e6ff2efd91d81dbc4d5e2edc13a35ba2" title="Click to view the MathML source">a,b>0$a,b>0$ are constants, mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039614001557&_mathId=si4.gif&_user=111111111&_pii=S0022039614001557&_rdoc=1&_issn=00220396&md5=1206c02bc2044e3415d392c45b132608" title="Click to view the MathML source">2<p<5$2<p<5$ and mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039614001557&_mathId=si5.gif&_user=111111111&_pii=S0022039614001557&_rdoc=1&_issn=00220396&md5=ee7d1e56af89551db879e8c99d988eb0" title="Click to view the MathML source">V:R³→R. Under certain assumptions on V, we prove that (0.1) has a positive ground state solution by using a monotonicity trick and a new version of global compactness lemma.

Our main results especially solve problem (0.1) in the case where mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039614001557&_mathId=si6.gif&_user=111111111&_pii=S0022039614001557&_rdoc=1&_issn=00220396&md5=1d99826a0f087acccd280082b69e4197" title="Click to view the MathML source">p∈(2,3], which has been an open problem for Kirchhoff equations and can be viewed as a partial extension of a recent result of He and Zou in [14] concerning the existence of positive solutions to the nonlinear Kirchhoff problem

where mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039614001557&_mathId=si8.gif&_user=111111111&_pii=S0022039614001557&_rdoc=1&_issn=00220396&md5=13679829384bf6e78527d9bbc8fd9085" title="Click to view the MathML source">蔚>0$蔚>0$ is a parameter, mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039614001557&_mathId=si9.gif&_user=111111111&_pii=S0022039614001557&_rdoc=1&_issn=00220396&md5=cf0ce6e6a1fc18801abc47ecf6718a15" title="Click to view the MathML source">V(x) is a positive continuous potential and mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039614001557&_mathId=si10.gif&_user=111111111&_pii=S0022039614001557&_rdoc=1&_issn=00220396&md5=c22335157a62039186e44fd5bb65c0ef" title="Click to view the MathML source">f(u)∼|u|^p−1u with mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039614001557&_mathId=si11.gif&_user=111111111&_pii=S0022039614001557&_rdoc=1&_issn=00220396&md5=1a08bb19cf70eae6e465c966f8dcf3bf" title="Click to view the MathML source">3<p<5$3<p<5$ and satisfies the Ambrosetti–Rabinowitz type condition. Our main results extend also the arguments used in [7] and [33], which deal with Schrödinger–Poisson system with pure power nonlinearities, to the Kirchhoff type problem.