2012/06/15 by Craig Cowan, Cowan, Craig, Nassif Ghoussoub +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations #math.AP
paper · pdf · doi:10.48550/arxiv.1206.3471
arXiv admin note: text overlap with arXiv:1003.3862
arxiv created 2012/06/15 · openalex publication_date 2012/06/15 · arxiv updated 2012/06/18 · openalex created_date 2022/09/28 · openalex updated_date 2026/07/28
We examine the fourth order problem Δ2 u = λf(u) in Ω with Δu = u =0 on ∂ Ω, where λ> 0 is a parameter, Ω is a bounded domain in RN and where f is one of the following nonlinearities: f(u)=eu, f(u)=(1+u)p or f(u)= (1)/((1-u)p) where p>1. We show the regularity of all semi-stable solutions and hence of the extremal solutions, provided [N < 2 + 4 √(2) + 4 √(2 - √(2)) ≈ 10.718 when f(u)=eu,] and [(N)/(4) < (p)/(p-1) + (p+1)/(p-1) (√((2p)/(p+1)) + √((2p)/(p+1) - √((2p)/(p+1))) - 1/2)] when f(u)=(u+1)p. New results are also obtained in the case where f(u)=(1-u)-p. These are substantial improvements to various results on critical dimensions obtained recently by various authors. We view the equation as a system and then derive a new stability inequality, valid for minimal solutions, which allows a method of proof which is reminiscent of the second order case.