A biharmonic eigenvalue problem and ITS application

In this article, we focus on the eigenvalue problem of the following linear biharmonic equation in ℝN△2u−αu+λg(x)u=0 withu∈H2(ℝN),u≠0, N≥5.△2u−αu+λg(x)u=0 withu∈H2(ℝN),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 for a suitable 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 ℝN.