Electron. J. Diff. Equ., Vol. 2013 (2013), No. 70, pp. 1-8.

Point rupture solutions of a singular elliptic equation

Huiqiang Jiang, Attou Miloua

Abstract:
We consider the elliptic equation
$$ \Delta u=f(u) $$
in a region $\Omega\subset\mathbb{R}^2$, where f is a positive continuous function satisfying
$$ \lim_{u\to 0^{+}}f(u) =\infty. $$
Motivated by the thin film equations, a solution $u$ is said to be a point rupture solution if for some $p\in\Omega$, $u(p) =0$ and $u(p) >0$ in $\Omega\backslash\{ p\} $. Our main result is a sufficient condition on f for the existence of radial point rupture solutions.

Submitted May 2, 2012. Published March 13, 2013.
Math Subject Classifications: 49Q20, 35J60, 35Q35.
Key Words: Thin film; point rupture; radial solution; singular equation.

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Huiqiang Jiang
Department of Mathematics, University of Pittsburgh
Pittsburgh, PA 15260, USA
email: hqjiang@pitt.edu
Attou Miloua
Department of Mathematics, University of Pittsburgh
Pittsburgh, PA 15260, USA
email: atm33@pitt.edu

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