Shooting Method with Sign-Changing Nonlinearity

TL;DR

This paper extends a shooting method with degree theory to prove the existence of solutions for nonlinear systems with sign-changing nonlinearities, broadening applicability to systems like nonlinear Schrödinger equations.

Contribution

It introduces new dynamic estimates that enable the degree theory approach to handle systems with sign-changing source terms, expanding previous methods.

Findings

Established existence results for systems with sign-changing nonlinearities

Extended the shooting method to broader classes of nonlinear systems

Provided new analytical tools for nonlinear PDEs with sign-changing sources

Abstract

In this paper, we study the existence of solution to a nonlinear system: \begin{align} \left\{\begin{array}{cl} -\Delta u_{i} = f_{i}(u) & \text{in } \mathbb{R}^n, u_{i} > 0 & \text{in } \mathbb{R}^n, \, i = 1, 2,\cdots, L % u_{i}(x) \rightarrow 0 & \text{uniformly as } |x| \rightarrow \infty \end{array} \right. \end{align} for sign changing nonlinearities $f_i$'s. Recently, a degree theory approach to shooting method for this broad class of problems is introduced in \cite{LiarXiv13} for nonnegative $f_i$'s. However, many systems of nonlinear Sch\"odinger type involve interaction with undetermined sign. Here, based on some new dynamic estimates, we are able to extend the degree theory approach to systems with sign-changing source terms.