Multiplicity Of Positive Solutions To Singular Superlinear Equations With Neumann Boundary Value Problems

To the knowledge of the authors, singular second-order differential equation boundary value problems have been studied extensively in the literature. Most papers are concerned with the case p(x) = - 1, q(x) = 0 and p(x) = -1, q(x) ≠ 0. However, for the case p(x) ≠ 1 and q(x) ≠ 0,the main results have not been improved and generalized in the literature.In this paper, we are devoted to establish the multiplicity of positive solutions to superlinear attractive singular equations with Neumann boundary conditions. It is proved that such a problem has at least two positive solutions under our reasonable conditions. Our nonlinearity may be singular in its dependent variable and superlinear at infinity. The proof relies on a nonlinear alternative of Leray-Schauder type and on Krasnoselskii fixed point theorem on compression and expansion of cones. The Green Function is also important in the proof.. The existence of the first solution is obtained using a nonlinear alternative of Leray-Schauder, and the second one is found using a fixed point theorem in cones.Besides fixed point theorems in a cone used in the existence problems, another tool- -the method of upper and lower solutions—is also used in the literature[l-5, 8, 13, 15]. In fact, the method of upper and lower solutions is much more frequently used.