Positive solutions to a fractional equation with singular nonlinearity

2017/06/06 by Jacques Giacomoni, Adimurthi, Giacomoni, Jacques +2
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1706.01965

openalex publication_date 2017/06/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the positive solutions to the following singular and non local elliptic problem posed in a bounded and smooth domain Ω⊂ \R^N, N> 2s: % (P_λ)\· amp;(-Δ)^s u=λ(K(x)u^-δ+f(u)) in Ω · amp;u · gt;0 in Ω · amp; u≡ 0 in \R^N\backslashΩ.. % Here 00, λ>0 and f : \R⁺→\R⁺ is a positive C² function. K : Ω→ \R⁺ is a Hölder continuous function in Ω which behave as \rm dist(x,∂Ω)^-β near the boundary with 0≤ β<2s. First, for any δ>0 and for λ> small enough, we prove the existence of solutions to (P_λ). Next, for a suitable range of values of δ, we show the existence of an unbounded connected branch of solutions to (P_λ) emanating from the trivial solution at λ=0. For a certain class of nonlinearities f, we derive a global multiplicity result that extends results proved in \citeperal-al. To establish the results, we prove new properties which are of independent interest and deal with the behavior and Hölder regularity of solutions to (P_λ).

Related

• Global branching for semilinear fractional Laplace with sublinear nonlinearity
• Positive solutions for semilinear fractional elliptic problems involving an inverse fractional operator
• Arbitrary many positive solutions for a nonlinear problem involving the fractional Laplacian
• Existence, Uniqueness of Positive Solution to a Fractional Laplacians with Singular Nonlinearity
• Boundary value problem with measures for fractional elliptic equations\n involving source nonlinearities