Spectral and topological methods in the study of solvability of semilinear equations in Hilbert spaces

TL;DR

This dissertation develops spectral and topological methods to establish conditions for the existence of solutions to semilinear equations in Hilbert spaces, especially when zero lies in the essential spectrum of the linear operator.

Contribution

It introduces novel approaches for handling the case where zero is in the essential spectrum, using degree theory and monotonicity methods, extending previous results.

Findings

Established solution existence conditions for semilinear equations with zero in the essential spectrum

Applied degree theory and monotonicity techniques to nonlinear operator equations

Proved solvability of stationary semilinear Schrödinger equations

Abstract

The main goal of this dissertation is to find conditions which will guarantee the existence of solutions in the Hilbert space $H$ of semilinear equation \[ L u+N(u)=h \] where $L$ is a linear and self-adjoint operator, $N$ a non-linear mapping and $h\in H$. In this project we concentrate on the case when $0$ belongs to the essential spectrum of operator $L$ which was not previously studied in this general setting. In chapter 2 we additionally assume that $0$ is the infimum of the essential spectrum of $L$. We apply the degree theory for densely defined mappings of class $(S_+)$ to the operator given by the left hand side of the equation. We assume that non-linear part $N$ is quasi-monotone and satisfies sublinear growth condition. Moreover, since $0$ can have non-trivial eigenspace, we make use of the so called recession functional connected with $L$ and $N$ which allows to control the behaviour of non-linear part on the kernel of $L$. In chapter 3 we allow the essential spectrum of $L$ to lay below zero. Our method is based on the observation that certain perturbation of operator $L$ is maximal monotone on the subspace of $H$ corresponding to non-negative part of the spectrum of $L$ and we assume that operator $N$ satisfies certain monotonicity condition as well. Next we explore the surjectivity properties of maximal monotone operators to show the existence of solutions to the class of perturbed equations. Finally with the help of recession functional and a growth conditions on $N$ we prove the solvability of the equation. The application to the solvability of stationary semilinear Schr\"{o}dinger equation is given.