Nonlinear Singular Eigenvalue Problems

We study a nonlinear eigenvalue problem driven by a general nonhomogeneous differential operator, involving a reaction term that is singular at x = 0 and becomes superlinear as x → +∞. Unlike the usual case in the literature, the singular term and the perturbation are not decoupled. By using variational methods in combination with truncation and comparison techniques, we establish a global existence and multiplicity theorem with respect to the parameter (eigenvalue) λ > 0. Additionally, we demonstrate the existence of a minimal positive solution u*λ and investigate the continuity and monotonicity properties of the map λ → u*λ.