2018/01/04 by Mousomi Bhakta, Bhakta, Mousomi, Phuoc‐Tai Nguyen +1
Computer Science · Mathematics · #35J20 #35J66 #35R06 #35R11 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1801.01544
openalex publication_date 2018/01/04 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28
We are concerned with positive solutions of equation (E) (-\Δ)s u=f(u)\nin a domain \Ω \⊂ \ℝN (N>2s), where s \∈\n(\(1)/(2),1) and f\∈ C\αloc(\ℝ) for some \α\n\∈(0,1). We establish a universal a priori estimate for positive solutions of\n(E), as well as for their gradients. Then for C2 bounded domain \Ω, we\nprove the existence of positive solutions of (E) with prescribed boundary value\n\ρ \ν, where \ρ>0 and \ν is a positive Radon measure on \∂\n\Ω with total mass 1, and discuss regularity property of the solutions.\nWhen f(u)=up, we demonstrate that there exists a critical exponent\nps:=\(N+s)/(N-s) in the following sense. If p\≥ ps, the problem does\nnot admit any positive solution with \ν being a Dirac mass. If p\∈(1,ps)\nthere exits a threshold value \ρ^*>0 such that for \ρ\∈ (0, \ρ^*],\nthe problem admits a positive solution and for \ρ>\ρ^*, no positive\nsolution exists. We also show that, for \ρ>0 small enough, the problem\nadmits at least two positive solutions.\n