Tornado solutions for semilinear elliptic equations in ℝ²: Applications

Abstract

Abstract. We show partial regularity of bounded positive solutions of some semilinear elliptic equations ∆u = f (u) in domains of R 2 . As a consequence, there exists a large variety of nonnegative singular solutions to these equations. These equations have previously been studied from the point of view of free boundary problems, where solutions additionally are stable for a variational problem, which we do not assume.

Tornado solutions for semilinear elliptic equations in ℝ²: regularity

“…We establish conditions for regularity in terms of the existence of "tornado sequences" of solutions. In the second paper [5], we apply the results presented here to prove regularity properties and existence of singular solutions to ∆u = f (u) when f (u) = g(u)u −α , with 0 ≤ C 1 < g(u) < C 2 , g continuous away from u = 0. We begin with the definition of tornado sequences.…”