We are concerned with the multiplicity of positive solutions for the singular superlinear and subcritical Schrödinger equation $$-\mathrm{\Delta }u+V(x)u=\lambda a(x)u^{-\gamma }+b(x)u^p\text{in }{ℝ}^N,$$ 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 $b\in L^{\infty }({ℝ}^N)$ may change its sign, $0<a\in L^{\frac{2}{1+\gamma }}({ℝ}^N)$, $V$ is a positive continuous function, $N\ge 3$ and $\lambda >0$ 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.

我们关注奇异超线性和亚临界薛定谔方程的正解的多重性$$-\mathrm{\Delta }你+五(X)你=\lambda \mathrm{一个}(X){你}^{-\gamma }+b(X){你}^p\text{在 }{ℝ}^{ñ},$$超出 [Y. Il'yasov，通过非线性瑞利商， Topol 论 Nehari 流形的极值。方法非线性肛门。49 (2017) 683–714]，当潜力$b\in {\mathrm{大号}}^{\infty }({ℝ}^{ñ})$可能会改变它的标志，$0<\mathrm{一个}\in {\mathrm{大号}}^{\frac{2}{1+\gamma }}({ℝ}^{ñ})$,$五$是一个正连续函数，$ñ\ge 3$和$\lambda >0$是一个实参数。主要困难来自能量泛函的不可微性以及 Nehari 集的连通分量边界的交点非空这一事实。我们通过探索该边界的拓扑结构来克服这些困难，以构建其边界具有空交集的非空集，并通过控制能级来最小化它们。