Publication details
Semilinear fractional elliptic equations with source term and boundary measure data
| Authors | |
|---|---|
| Year of publication | 2022 |
| Type | Article in Periodical |
| Magazine / Source | Pure and Applied Functional Analysis |
| MU Faculty or unit | |
| Citation | |
| Web | http://yokohamapublishers.jp/online2/oppafa/vol7/p863.html |
| Keywords | nonlocal; source terms; a priori estimate; singularity estimate; existence; boundary trace; boundary singularity; isolated singularity; fractional; minimal solution; mountain pass solution; gradient nonlinearity |
| Description | A notion of s-boundary trace recently introduced by Nguyen and Véron (Adv. Nonlinear Stud. 18, 237-267, 2018) is an efficient tool to study boundary value problems with measure data for fractional elliptic equations with an absorption nonlinearity. In this paper, we investigate a fractional equation with a source term $(-\Delta)^s u=f(u)$ in $\Omega$ with a prescribed s-boundary trace $\rho \nu$, where $\Omega$ is a $C^2$ bounded domain of $\mathbb{R}^N$ ($N>2s$), $s \in (\frac{1}{2},1)$, $f\in C^{\beta}_{loc}(\mathbb{R})$, for some $\beta \in(0,1)$, $\nu$ is a positive Radon measure on $\partial \Omega$ with total mass 1 and $\rho$ is a positive parameter. We provide an existence result for the above equation and discuss regularity property of solutions. When $f(u)=u^p$, we prove that there exists a critical exponent $p_s:=\frac{N+s}{N-s}$ in the following sense. If $p\geq p_s$, the problem does not admit any positive solution with $\nu$ being a Dirac mass. If $p\in(1,p_s)$ there exits a threshold value $\rho^*>0$ such that for $\rho\in (0, \rho^*]$, the problem admits a positive solution and for $\rho>\rho^*$, no positive solution exists. We also show that, for $\rho>0$ small enough, the problem admits at least two positive solutions. |