The existence of solutions of Schr\"odinger equations with essence resonance

TL;DR

This paper studies solutions to asymptotically linear Schrödinger equations with spectrum resonance, introducing a bootstrap iteration method and spectrum comparison to establish existence and nondegeneracy of solutions without mountain pass geometry.

Contribution

It develops a novel bootstrap iteration approach for elliptic equations on RN and provides explicit conditions for solution existence and nondegeneracy in resonance cases.

Findings

Established existence of solutions without mountain pass geometry.

Introduced a bootstrap iteration method generalizing Agmon-Douglis-Nirenberg theorem.

Provided spectrum comparison results for Schrödinger operators.

Abstract

The current paper investigates a class of asymptotically linear Schrodinger equations. The Palais-Smale condition fails to hold in this case. Especially under the hypothesis (V2), 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 RN. The iteration is self-contained and can be regarded as a generalization of Agmon-Douglis-Nirenberg theorem. The proof characterizes iteration steps independent of the choice of the parameter, 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 in RN. Additionally, a comparison theorem for the spectrum of Schrodinger 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.