Existence of singular positive solutions for some semilinear elliptic equations

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 {u(alpha)}(alpha>0) with u(alpha)(0) = alpha and each of them is oscillatory in ( 0, infinity). We obtain then a singular radial solution in ( 0, infinity) by taking the limit alpha --> 0. Meanwhile, using the solutions obtained in ( 0, infinity), we show some existence results for the corresponding Neumann eigenvalue problem on a ball.