Some Remarks on Biharmonic Elliptic Problems with a Singular Nonlinearity

Abstract EN

We study the following semilinear biharmonic equation Δ 2 u = λ / 1 - u , in ? , and u = ∂ u / ∂ n = 0 , on ∂ ? , where ? is the unit ball in ℝ n 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 u ̲ λ , which satisfies 0 < u ̲ λ < 1 .

In the extremal case λ = λ * , we prove the existence of a weak solution which is the 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.