Semilinear fractional elliptic equations with source term and boundary measure data

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216224%3A14310%2F22%3A00128818" target="_blank" >RIV/00216224:14310/22:00128818 - isvavai.cz</a>
Result on the web
<a href="http://yokohamapublishers.jp/online2/oppafa/vol7/p863.html" target="_blank" >http://yokohamapublishers.jp/online2/oppafa/vol7/p863.html</a>
DOI - Digital Object Identifier
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Result language
angličtina
Original language name
Semilinear fractional elliptic equations with source term and boundary measure data
Original language 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 $mathbb{R}^N$ ($N>2s$), $s in (frac{1}{2},1)$, $fin C^{beta}_{loc}(mathbb{R})$, 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 $pgeq p_s$, the problem does not admit any positive solution with $nu$ being a Dirac mass. If $pin(1,p_s)$ there exits a threshold value $rho^*>0$ such that for $rhoin (0, rho^*]$, 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.
Czech name
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Czech description
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Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2022
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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