2018/01/04 by Mousomi Bhakta, Bhakta, Mousomi, Phuoc‐Tai Nguyen +2
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 #math.AP #msc:35J20 #msc:35J66 #msc:35R06 #msc:35R11
paper · pdf · doi:10.48550/arxiv.1801.01544
We withdraw the paper because there was a flaw in the proof of Theorem 3.1 in Section 3. As a consequence, Theorem 1.2, Theorem 3.4 and Theorem A.1 are not valid
openalex publication_date 2018/01/04 · arxiv created 2020/09/29 · arxiv updated 2020/09/30 · 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) in a domain Ω⊂ ℝN (N>2s), where s ∈ ((1)/(2),1) and f∈ Cαloc(ℝ) for some α∈(0,1). We establish a universal a priori estimate for positive solutions of (E), as well as for their gradients. Then for C2 bounded domain Ω, we prove the existence of positive solutions of (E) with prescribed boundary value ρν, where ρ>0 and ν is a positive Radon measure on ∂ Ω with total mass 1, and discuss regularity property of the solutions. When f(u)=up, we demonstrate that there exists a critical exponent ps:=(N+s)/(N-s) in the following sense. If p≥ ps, the problem does not admit any positive solution with ν being a Dirac mass. If p∈(1,ps) there exits a threshold value ρ^*>0 such that for ρ∈ (0, ρ^*], the problem admits a positive solution and for ρ>ρ^*, no positive solution exists. We also show that, for ρ>0 small enough, the problem admits at least two positive solutions.