On the critical dimension of a fourth order elliptic problem with negative exponent

2009/05/12 by Amir Moradifam, Moradifam, Amir
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations #math.AP

paper · pdf · doi:10.48550/arxiv.0905.1940

arxiv created 2009/05/12 · openalex publication_date 2009/05/12 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the regularity of the extremal solution of the semilinear biharmonic equation βΔ² u-τΔu=\fracλ(1-u)² on a ball B ⊂ \R^N, under Navier boundary conditions u=Δu=0 on ∂ B, where λ>0 is a parameter, while τ>0, β>0 are fixed constants. It is known that there exists a λ^* such that for λ>λ^* there is no solution while for λ<λ^* there is a branch of minimal solutions. Our main result asserts that the extremal solution u^* is regular (sup_Bu^*<1) for N≤ 8 and β, τ>0 and it is singular (sup_Bu^*=1) for N≥ 9, β>0, and τ>0 with \fracτβ small. Our proof for the singularity of extremal solutions in dimensions N≥ 9 is based on certain improved Hardy-Rellich inequalities.

Citations

• Regularity of the extremal solution in a MEMS model with advection

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