Existence and Uniqueness of Positive Solution for a Fractional Dirichlet Problem with Combined Nonlinear Effects in Bounded Domains

Abstract

We prove the existence and uniqueness of a positive continuous solution to the following singular semilinear fractional Dirichlet problem ${(-\mathrm{\Delta })}^{\alpha /\mathrm{2}}u=a_{\mathrm{1}}(x)u^{{\sigma }_{\mathrm{1}}}+a_{\mathrm{2}}(x)u^{{\sigma }_{\mathrm{2}}}$, in $\mathrm{\hspace{0.17em}D\hspace{0.17em}}{\text{\hspace{0.17em}lim}}_{x\to z\in \partial D}(\delta (x){)}^{\mathrm{1}-(\mathrm{\alpha }/\mathrm{2})}u(x)=0,$ where ${\sigma }_{\mathrm{1}},{\sigma }_{\mathrm{2}}\in (-\mathrm{1,1}),\hspace{0.17em}$ $D$ is a bounded $C^{\mathrm{1,1}}$-domain in ${ℝ}^n,$ $n\ge \mathrm{2},$ and $\delta (x)$ denotes the Euclidian distance from $x$ to the boundary of $D.$ The nonnegative weight functions $a_{\mathrm{1}},a_{\mathrm{2}}$ are required to satisfy certain hypotheses related to the Karamata class. We also investigate the global behavior of such solution.