2008/01/16 by Juan Dávila, Louis Dupaigne, Davila, Juan +5
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics
paper · doi:10.48550/arxiv.0801.2445
openalex publication_date 2008/01/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let λ^*>0 denote the largest possible value of λ such that \\beginaligned Δ2 u amp; = \la eu amp;amp; in B u amp;= \pdun = 0 amp;amp; on \pa B \endaligned . has a solution, where B is the unit ball in \RN and n is the exterior unit normal vector. We show that for λ=λ^* this problem possesses a unique \em weak solution u^*. We prove that u^* is smooth if N≤ 12 and singular when N≥ 13, in which case u^*(r) = - 4 log r + log (8(N-2)(N-4) / λ^*) + o(1) as r→ 0. We also consider the problem with general constant Dirichlet boundary conditions.