Tornado solutions for semilinear elliptic equations in ℝ²: Applications

Hammock, Frances; Luthy, Peter; Meadows, Alexander; Whitman, Phillip

doi:10.1090/S0002-9939-06-08618-7

Tornado solutions for semilinear elliptic equations in $\mathbb {R}^2$: Applications
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by Frances Hammock, Peter Luthy, Alexander M. Meadows and Phillip Whitman PDF

Proc. Amer. Math. Soc. 135 (2007), 1419-1430 Request permission

Abstract:

We show partial regularity of bounded positive solutions of some semilinear elliptic equations $\Delta u = f(u)$ in domains of $\mathbb {R}^2$. 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.

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