Vol. 236, No. 1, 2008

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Existence of singular positive solutions for some semilinear elliptic equations

Zongming Guo, Dong Ye and Feng Zhou

Vol. 236 (2008), No. 1, 57–71
Abstract

We study positive solutions of an equation with singular nonlinearities. The equation arises in the study of equilibrium states of thin films. Under weak assumptions on the nonlinearity, we show that for N 3 there exists a family of radial solutions {uα}α>0 with uα(0) = α and each of them is oscillatory in (0,). We obtain then a singular radial solution in (0,) by taking the limit α 0. Meanwhile, using the solutions obtained in (0,), we show some existence results for the corresponding Neumann eigenvalue problem on a ball.

Keywords
elliptic equation with singular nonlinearity, rupture solutions, oscillation
Mathematical Subject Classification 2000
Primary: 35J60, 35B45, 35B05
Milestones
Received: 11 May 2007
Revised: 27 September 2007
Accepted: 25 February 2008
Published: 1 May 2008
Authors
Zongming Guo
Department of Mathematics
Henan Normal University
Xinxiang 453002
China
Dong Ye
Département de Mathématiques, UMR 8088
Université de Cergy-Pontoise
Site Saint-Martin, BP 222
95302 Cergy-Pontoise Cedex
France
Feng Zhou
Department of Mathematics
East China Normal University
Shanghai 200062
China