On the number of solutions of a quadratic equation in a normed space

TL;DR

This paper investigates the solutions of quadratic equations in normed spaces, establishing conditions for uniqueness and applying these results to a PDE boundary value problem involving the Laplacian.

Contribution

It introduces conditions under which quadratic equations in normed spaces have only symmetric solutions and applies these to PDE boundary value problems.

Findings

Conditions for solution uniqueness in quadratic equations

Application to Dirichlet boundary value problems

Insights into symmetry of solutions

Abstract

We study an equation $Qu=g$, where $Q$ is a continuous quadratic operator acting from one normed space to another normed space. Obviously, if $u$ is a solution of such equation then $-u$ is also a solution. We find conditions implying that there are no other solutions and apply them to the study of the Dirichlet boundary value problem for the partial differential equation $u\Delta u =g$.