Blow up boundary solutions of some semilinear fractional equations in the unit ball (Q276734)

The authors are concerned with the study of nonnegative solutions \(u\) to \(\Delta ^{\alpha/2}u=u^\gamma\) in \(B\), \(u=0\) in \(B^c\), that blow up at the boundary \(\partial B\). Here \(B\) is the unit ball in \({\mathbb R}^N\), \(N\geq 3\), \(\alpha\in (0,2)\). In their study, the authors identify two types of solutions in terms of their behavior at the boundary. Firstly, the solutions blowing up at \(\partial B\) with the rate \(\text{dist}(x,\partial B)^{\alpha/2-1}\) are observed. Secondly, the solutions growing faster than \(\text{dist}(x,\partial B)^{\alpha/2-1}\) are considered. In particular, no Keller-Osserman condition exists as in the classical case \(\alpha=2\).