A BIHARMONIC EIGENVALUE PROBLEM AND ITS APPLICATION

In this article, we focus on the eigenvalue problem of the following linear biharmonic equation in R^N： △^2u-αu＋λg（x）u=0 with u ∈H^2（R^N）,u≠0,N≥5 Note that there are two parameters α and λ in it, which is different from the usual eigenvalue problems. Here, we consider λ as an eigenvalue and seek Ior...

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Bibliographic Details
Published in:	Acta mathematica scientia Vol. 32; no. 3; pp. 1213 - 1225
Main Author:	王江潮张贻民
Format:	Journal Article
Language:	English
Published:	Elsevier Ltd 01-05-2012 Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China
Subjects:	35J20 35J60 asymptotically linear Biharmonic equation Biharmonic equations eigenvalue problem Eigenvalues IOR potential well 双调和方程应用损失最大值原理渐近线性特征值问题非平凡解 potential well Biharmonic equation 35J20 asymptotically linear 35J60 eigenvalue problem
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Summary:	In this article, we focus on the eigenvalue problem of the following linear biharmonic equation in R^N： △^2u-αu＋λg（x）u=0 with u ∈H^2（R^N）,u≠0,N≥5 Note that there are two parameters α and λ in it, which is different from the usual eigenvalue problems. Here, we consider λ as an eigenvalue and seek Ior a sulble range of parameter α, which ensures that problem （＊） has a maximal eigenvalue. As the loss of strong maximum principle for our problem, we can only get the existence of non-trivial solutions, not positive solutions, in this article. As an application, by using these results, we studied also the existence of non-trivial solutions for an asymptotically linear biharmonic equation in R^N.
Bibliography:	In this article, we focus on the eigenvalue problem of the following linear biharmonic equation in R^N： △^2u-αu＋λg（x）u=0 with u ∈H^2（R^N）,u≠0,N≥5 Note that there are two parameters α and λ in it, which is different from the usual eigenvalue problems. Here, we consider λ as an eigenvalue and seek Ior a sulble range of parameter α, which ensures that problem （＊） has a maximal eigenvalue. As the loss of strong maximum principle for our problem, we can only get the existence of non-trivial solutions, not positive solutions, in this article. As an application, by using these results, we studied also the existence of non-trivial solutions for an asymptotically linear biharmonic equation in R^N. 42-1227/O Biharmonic equation; potential well; eigenvalue problem; asymptotically linear ObjectType-Article-2 SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 23
ISSN:	0252-9602 1572-9087
DOI:	10.1016/S0252-9602(12)60093-9