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 −Δu + V (x)u = λa(x)u−γ + b(x)upin ℝN, beyond the Nehari extremal value, as defined in [Y. Il’yasov, On extreme values of Nehari manifold via nonlinear Rayleigh’s quotient, <italic>Topol. Methods Nonlinear Anal.</italic> <bold>49</bold> (2017) 683–714], when the potential b ∈ L∞(ℝN) may change its sign, 0 < a ∈ L 2 1+γ(ℝN), V is a positive continuous function, N ≥ 3 and λ > 0 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. [ABSTRACT FROM AUTHOR]