Existence and uniqueness of positive solution for a fractional Dirichlet problem with combined nonlinear effects in bounded domains
Summary: We prove the existence and uniqueness of a positive continuous solution to the following singular semilinear fractional Dirichlet problem \((-\Delta)^{\alpha/2}u=a_1(x)u^{\sigma_1}+a_2(x)u^{\sigma_2}\) in \(D\), \(\lim_{x\to z\in\partial D}(\delta(x))^{1-(\alpha/2)}u(x)=0\), 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 \(a_1\), \(a_2\) are required to satisfy certain hypotheses related to the Karamata class. We also investigate the global behavior of such solutions.
scientific article; zbMATH DE number 6306629
scientific article