Solvability of semilinear equations with zero on the boundary of spectral gap and applications to nonlinear Schr\"{o}dinger equation

TL;DR

This paper investigates the existence of solutions to semilinear equations involving self-adjoint operators with spectral gaps, applying the results to nonlinear stationary Schrödinger equations in Hilbert spaces.

Contribution

It provides new solvability conditions for semilinear equations with spectral gap boundary points, specifically addressing cases where zero lies in the essential spectrum.

Findings

Established sufficient conditions for solution existence based on operator monotonicity and sign assumptions.

Applied the theoretical results to nonlinear stationary Schrödinger equations in Euclidean space.

Demonstrated the role of spectral gap boundary points in the solvability of semilinear equations.

Abstract

We study the existence of solutions in Hilbert space $H$ of the semilinear equation \[ L u+N(u)=h, \] where $L$ is linear self-adjoint, $N$ is a nonlinear operator and $h\in H$. We concentrate on the case when $0$ is a right boundary point of a gap in the spectrum of $L$ and an element of essential spectrum. The sufficient conditions for solvability are based on monotonicity and sign assumptions on operator $N$, and its behaviour on $\ker L$. We illustrate the main theorem by an application to the study of nonlinear stationary Schr\"{o}dinger equation on $\mathbb{R}^n$.