Semilinear fractional elliptic equations involving measures

Abstract

We study the existence of weak solutions of (E) $(-\Delta {)}^{\alpha }u+g(u)=\nu$ in a bounded regular domain $\Omega$ in ${\mathbb{R}}^N(N\ge 2)$ which vanish on ${\mathbb{R}}^N\setminus \Omega$, where $(-\Delta {)}^{\alpha }$ denotes the fractional Laplacian with $\alpha \in (0,1)$, $\nu$ is a Radon measure and $g$ is a nondecreasing function satisfying some extra hypothesis. When $g$ satisfies a subcritical integrability condition, we prove the existence and uniqueness of a weak solution for problem (E) for any measure. In the case where $\nu$ is Dirac measure, we characterize the asymptotic behavior of the solution. When $g(r)=|r{|}^{k-1}r$ with $k$ supercritical, we show that a condition of absolute continuity of the measure with respect to some Bessel capacity is a necessary and sufficient condition in order (E) to be solved.

References in corpus (6)

• Hitchhiker's guide to the fractional Sobolev spaces
• Existence and stability of solutions of general semilinear elliptic equations with measure data
• Singular solutions of fractional elliptic equations with absorption
• Variational problems related to some fractional kinetic equations
• The Dirichlet problem for the fractional Laplacian: regularity up to the boundary
• Quasilinear Lane-Emden equations with absorption and measure data