Abstract
We study the following semilinear biharmonic equation: 
, where 0 ≤ f ≤ 1 and BR ⊂ ℝN, N ≥ 1, is the ball centered in the origin of radius R. We prove, under Dirichlet boundary conditions u = Әu/Әƞ = 0 on ӘBR, the existence of λ* = λ*(R, f) > 0 such that for λ ∈ (0, λ*) there exists a minimal (classical) solution u̲λ < which satisfies 0 < u̲λ < 1. In the extremal case λ = λ*, we prove the existence of a weak solution which has finite energy and which is the unique solution even in a very weak sense. For λ > λ* there are no solutions of any kind. Estimates on λ*, stability properties of solutions and nonexistence results in the whole space are also established.
Keywords: Biharmonic operator; semilinear elliptic equations; singular nonlinearity; minimal solutions; extremal solutions; nonlinear eigenvalue problems; MEMS devices
Received: 2008-05-05
Published Online: 2016-03-10
Published in Print: 2009-02-01
© 2016 by Advanced Nonlinear Studies, Inc.
