Differential and Integral Equations

Combined effects in nonlinear singular fractional Dirichlet problem in bounded domains

Imed Bachar and Habib Mâagli

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Abstract

This paper deals with the existence and uniqueness of a positive continuous solution to the following singular semilinear fractional Dirichlet problem: \begin{equation*} \left( -\Delta _{\mid D}\right) ^{\frac{\alpha }{2}}u=a_{1}(x)u^{\sigma _{1}}+a_{2}(x)u^{\sigma _{2}}\text{ in }D,\text{ }\underset{x\rightarrow z\in \partial D}{\lim }\left( \delta (x)\right) ^{2-\alpha }u(x)=0, \end{equation*} where $0 < \alpha < 2,$ $\sigma _{1},\sigma _{2}\in (-1,1),$ $D$ is a bounded $C^{1,1}$-domain in $\mathbb{R}^{n},$ $n\geq 2,$ and $\delta (x)$ denotes the Euclidian distance from $x$ to the boundary of $D$. The nonnegative weight functions $a_{1}$ and $a_{2}$ are in $C_{loc}^{\gamma }(D),$ $ 0 < \gamma < 1,$ satisfying some appropriate assumptions related to Karamata regular variation theory. We also give the global behavior of such a solution.

Article information

Source
Differential Integral Equations, Volume 26, Number 11/12 (2013), 1361-1378.

Dates
First available in Project Euclid: 4 September 2013

Permanent link to this document
https://projecteuclid.org/euclid.die/1378327430

Mathematical Reviews number (MathSciNet)
MR3129013

Zentralblatt MATH identifier
1313.35358

Subjects
Primary: 34A08: Fractional differential equations 34B27: Green functions 35B09: Positive solutions

Citation

Bachar, Imed; Mâagli, Habib. Combined effects in nonlinear singular fractional Dirichlet problem in bounded domains. Differential Integral Equations 26 (2013), no. 11/12, 1361--1378. https://projecteuclid.org/euclid.die/1378327430


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