The current paper investigates a class of asymptotically linear Schrödinger equations with the nonlinear term \(f\in C\left( \mathbb {R},\mathbb {R} \right) \) s.t. \(\lim \limits _{\left| t\right| \rightarrow \infty }\frac{ f\left( t\right) }{t}=\sigma _0\) , where \(\sigma _0\) stands for the threshold of essential spectrum of Schrödinger operator. Clearly the Palais-Smale condition fails to hold in this case. Especially under the hypothesis \( \left( V_2\right) \) , the lack of compactness occurs at the interaction between nonlinear term and continuum spectrum. For this reason, we introduce a bootstrap iteration approach for elliptic equation on \(\mathbb {R}^N\) . The iteration is self-contained and can be regarded as a generalization of Agmon-Douglis-Nirenberg theorem (see Agmon et al. (Comm Pure Appl Math 17:35—92, 1964), Agmon et al. (Comm Pure Appl Math 12:623—727, 1959)). The proof characterizes iteration steps independent of the choice of the parameter \( \lambda \) , which are indeed manipulated by intrinsic natures of potentials and nonlinear terms, and furthermore presents precise estimates for asymptotically linear functions or continuous nonlinear terms restricted on a bounded domain \(\Omega \) with smooth boundary \(\partial \Omega \) in \(\mathbb {R }^N\) . Additionally, a comparison theorem for the spectrum of Schrödinger operator is also established in this paper. With above preparations, we can get a nontrivial solution without mountain pass geometry, and more importantly make an explicit description of nondegeneracy of solutions with monotonicity hypothesis.
