Existence, Nonexistence and Multiplicity of Positive Solutions for Generalized Laplacian Problems with a Parameter

Abstract

We investigate the homogeneous Dirichlet boundary value problem for generalized Laplacian equations with a singular, potentially non-integrable weight. By examining asymptotic behaviors of the nonlinear term near 0 and ∞, we establish the existence, nonexistence, and multiplicity of positive solutions for all positive values of the parameter λ. Our proofs rely on the fixed point theorem concerning cone expansion and compression of norm type and the Leray–Schauder’s fixed point theorem.