Boundary Singularities of Solutions to Semilinear Fractional Equations

Phuoc-Tai Nguyen,Laurent Véron

doi:10.1515/ans-2017-6048

Abstract

Abstract We prove the existence of a solution of ( - Δ ) s ⁢ u + f ⁢ ( u ) = 0 {(-\Delta)^{s}u+f(u)=0} in a smooth bounded domain Ω with a prescribed boundary value μ in the class of Radon measures for a large class of continuous functions f satisfying a weak singularity condition expressed under an integral form. We study the existence of a boundary trace for positive moderate solutions. In the particular case where f ⁢ ( u ) = u p {f(u)=u^{p}} and μ is a Dirac mass, we show the existence of several critical exponents p. We also demonstrate the existence of several types of separable solutions of the equation ( - Δ ) s ⁢ u + u p = 0 {(-\Delta)^{s}u+u^{p}=0} in ℝ + N {\mathbb{R}^{N}_{+}} .

Highlights

We prove the existence of a solution of (−∆)su + f (u) = 0 in a smooth bounded domain Ω with a prescribed boundary value μ in the class of Radon measures for a large class of continuous functions f satisfying a weak singularity condition expressed under an integral form
We study boundary singularity problem for semilinear fractional equation of the form
Few papers concerning boundary singularity problem for nonlinear fractional elliptic equation have been published in the literature

Summary

Assume s

In measures framework, because of the interplay between the nonlocal operator (−∆)s and the nonlinearity term f (u), the analysis is much more intricate and there are 3 critical exponents p∗1 This yields substantial new difficulties and leads to disclose new types of results. For every τ ∈ M(Ω, ρs) and μ ∈ M(∂Ω) there exists a unique weak solution of (1.2). Let u and un be the unique weak solutions of (1.2) with data (τ, μ) and (τn, μn) respectively. Under the assumption of Theorem C, let z ∈ ∂Ω, k > 0 and uΩz,k be the unique weak solution of (−∆)su + f (u) = 0 in Ω tr s(u) = kδz (1.12). Let 0 < p < p∗2 and denote by uΩk the unique weak solution of (−∆)su + up = 0. II- If p∗1 < p < p∗2 there exists a unique positive solution ω∗ ∈ W0s,2(S+N−1) of (1.16). In Appendix, we discuss separable solutions of (1.15) and demonstrate Theorem E

Linear problems

Nonlinear problems