Multiplicity of negative-energy solutions for singular-superlinear Schrödinger equations with indefinite-sign potential

We are concerned with the multiplicity of positive solutions for the singular superlinear and subcritical Schrödinger equation [Formula: see text] beyond the Nehari extremal value, as defined in [Y. Il’yasov, On extreme values of Nehari manifold via nonlinear Rayleigh’s quotient, Topol. Methods Nonlinear Anal. 49 (2017) 683–714], when the potential [Formula: see text] may change its sign, [Formula: see text], [Formula: see text] is a positive continuous function, [Formula: see text] and [Formula: see text] is a real parameter. The main difficulties come from the non-differentiability of the energy functional and the fact that the intersection of the boundaries of the connected components of the Nehari set is non-empty. We overcome these difficulties by exploring topological structures of that boundary to build non-empty sets whose boundaries have empty intersection and minimizing over them by controlling the energy level.