Strongly singular nonhomogeneous eigenvalue problems

We consider a nonlinear Dirichlet problem driven by the sum of a p-Laplacian and of a q-Laplacian, 1<p<q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p<q$$\end{document}, (a (p, q)-equation). The reaction is parametric (eigenvalue problem) and exhibits the competing effects of a strongly singular term and of (p-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(p-1)$$\end{document}-superlinear Carathéodory perturbation. We show that when the parameter (eigenvalue) is small, then the problem has at least two positive bounded solutions which are bounded away from zero on compact sets.