2017/02/20 by Craig Cowan, Cowan, Craig, Abbas Moameni +1
Computer Science · Mathematics · Medicine · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Spondyloarthritis Studies and Treatments
paper · pdf · doi:10.48550/arxiv.1702.06034
openalex publication_date 2017/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Utilizing a new variational principle that allows dealing with problems beyond the usual locally compactness structure, we study problems with a supercritical nonlinearity of the type -Δu + u= a(x) f(u) in Ω with ∂νu=0 on ∂ Ω. Here Ω is a bounded domain with certain symmetry assumptions. We find positive nontrivial solutions in the case of suitable supercritical nonlinearities f by finding critical points of I where I(u)=∫Ω\ a(x) F^* ( (-Δu + u)/(a(x)) ) - a(x) F(u) \ dx, over the closed convex cone Km of nonnegative, symmetric and monotonic functions in H1(Ω) where F'=f and where F^* is the Fenchel dual of F. We mention two important comments: firstly that there is a hidden symmetry in the functional I due to the presence of a convex function and its Fenchel dual that makes it ideal to deal with super-critical problems lacking the necessary compactness requirement. Secondly the energy I is not at all related to the classical Euler-Lagrange energy associated with equation. After we have proven the existence of critical points u of I on Km we then unitize a new abstract variational approach (developed by one of the present authors in \citeMo,Mo2) to show these critical points in fact satisfy -Δu + u = a(x) f(u). In the particular case of f(u)=|u|p-2 u we show the existence of positive nontrivial solutions beyond the usual Sobolev critical exponent.