Existence of singular positive solutions for some semilinear elliptic equations

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

Primary: 35J60, 35B45, 35B05