Existence and convergence of solutions for nonlinear biharmonic equations on graphs

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:268

Article

Han, Xiaoli¹  Shao, Mengqiu¹  Zhao, Liang² 

[1] Tsinghua Univ, Dept Math Sci, Beijing 100084, Peoples R China

[2] Beijing Normal Univ, Sch Math Sci, Lab Math & Complex Syst MOE, Beijing 100875, Peoples R China

关键词: Sobolev space;    Biharmonic equation;    Locally finite graph;    Ground state;   

DOI  :  10.1016/j.jde.2019.10.007

来源: Elsevier

【 摘 要 】

In this paper, we first prove some propositions of Sobolev spaces defined on a locally finite graph G = (V, E), which are fundamental when dealing with equations on graphs under the variational framework. Then we consider a nonlinear biharmonic equation Lambda(2)u - Lambda u + (lambda a + 1)u = vertical bar u vertical bar(p-2)u on G = (V, E). Under some suitable assumptions, we prove that for any lambda > 1 and p > 2, the equation admits a ground state solution u(lambda). Moreover, we prove that as lambda -> +infinity, the solutions u(lambda) converge to a solution of the equation {Delta(2)u - Delta u + u = vertical bar u vertical bar(p-2)u, in Omega, u=0, on partial derivative Omega, where Omega ={x is an element of V: a(x) = 0} is the potential well and partial derivative Omega denotes theboundary of Omega. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

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