N-Laplacian Equation with a Nonlinear Neumann Boundary Condition and a Singular Nonlinearity

Abstract: We study the existence, nonexistence, and multiplicity of a weak solution of the singular Neumann problem (Formula Presented.) where (Formula Presented.) , is a bounded smooth domain, (Formula Presented.) is the N -Laplace operator, (Formula Presented.) , and (Formula Presented.) is a constant. Here (Formula Presented.) function having superlinear growth at infinity and (Formula Presented.) . Using the sub-supersolution method and the variational method, under appropriate assumptions on g and h, we show that there exists a domain (Formula Presented.) bounded by the graph of a map (Formula Presented.) such that (Formula Presented.) admits at least two solutions for all (Formula Presented.) at least one solution for (Formula Presented.) , and no solution for all (Formula Presented.) .