Existence, Non-existence, Uniqueness of solutions for semilinear elliptic equations involving measures concentrated on boundary

Abstract

The purpose of this paper is to study the weak solutions of the fractional elliptic problem \begin{equation}\label{000} \begin{array}{lll} (-Δ)^αu+εg(u)=k\frac{\partial^αν}{\partial \vec{n}^α}\quad &{\rm in}\quad\ \ \barΩ,\\[3mm] \phantom{(-Δ)^α+εg(u)} u=0\quad &{\rm in}\quad\ \ \barΩ^c, \end{array} \end{equation} where $k>0$, $\epsilon =1$ or $-1$, $(-\Delta {)}^{\alpha }$ with $\alpha \in (0,1)$ is the fractional Laplacian defined in the principle value sense, $\Omega$ is a bounded $C^2$ open set in $R^N$ with $N\ge 2$, $\nu$ is a bounded Radon measure supported in $\partial \Omega$ and $\frac{{\partial }^{\alpha }\nu }{\partial {\vec{n}}^{\alpha }}$ is defined in the distribution sense, i.e. $$⟨\frac{{\partial }^{\alpha }\nu }{\partial {\vec{n}}^{\alpha }},\zeta ⟩={\int }_{\partial \Omega }\frac{{\partial }^{\alpha }\zeta (x)}{\partial {\vec{n}}_x^{\alpha }}d\nu (x),\forall \zeta \in C^{\alpha }(R^N),$$ here ${\vec{n}}_x$ denotes the unit inward normal vector at $x\in \partial \Omega$.

38 pages

References in corpus (1)

• Elliptic Equations Involving Meausres

Cited by in corpus (2)

• Boundary blow-up solutions to fractional elliptic equations in a measure framework
• On a class of semilinear fractional elliptic equations involving outside Dirac data