Some remarks on biharmonic elliptic problems with a singular nonlinearity

2011/01/20 by Baishun Lai, Lai, Baishun
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Composite Material Mechanics #Nonlinear Partial Differential Equations #math.AP

paper · pdf · doi:10.48550/arxiv.1101.3904

19 pages

arxiv created 2011/01/20 · arxiv updated 2011/01/21

Abstract

We study the following semilinear biharmonic equation \Δ²u=\fracλ1-u, · in \B, u=(∂ u)/(∂ n)=0, · on ∂\B, . %\eqno(M_λ) where \B is the unit ball in \Rⁿ and n is the exterior unit normal vector. We prove the existence of λ^*>0 such that for λ∈ (0,λ^*) there exists a minimal (classical) solution \underlineu_λ, which satisfies 0<\underlineu_λ<1. In the extremal case λ=λ^*, we prove the existence of a weak solution which is unique solution even in a very weak sense. Besides, several new difficulties arise and many problems still remain to be solved. we list those of particular interest in the final section.

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