Large $s$-harmonic functions and boundary blow-up solutions for the fractional Laplacian

Abstract

We present a notion of weak solution for the Dirichlet problem driven by the fractional Laplacian, following the Stampacchia theory. Then, we study semilinear problems of the form $$ \left\lbrace\begin{array}{ll} (-\triangle)^s u = \pm\,f(x,u) & \hbox{ in }\Omega \\ u=g & \hbox{ in }\mathbb{R}^n\setminus\overline{\Omega}\\ Eu=h & \hbox{ on }\partial\Omega \end{array}\right. $$ when the nonlinearity $f$ and the boundary data $g,h$ are positive, but allowing the right-hand side to be both positive or negative and looking for solutions that blow up at the boundary. The operator $E$ is a weighted limit to the boundary: for example, if $\Omega$ is the ball $B$, there exists a constant $C(n,s)>0$ such that $$ Eu(\theta) = C(n,s) \lim_{x \to \theta}_{x\in B} u(x) {dist(x,\partial B)}^{1-s}, \hbox{ for all } \theta \in \partial B. $$ Our starting observation is the existence of $s$-harmonic functions which explode at the boundary: these will be used both as supersolutions in the case of negative right-hand side and as subsolutions in the positive case.