Symmetry breaking and Morse index of solutions of nonlinear elliptic problems in the plane

TL;DR

This paper analyzes the symmetry and stability of solutions to a nonlinear elliptic PDE in the plane, computing Morse indices and establishing bifurcation of nonradial solutions, especially for exponential nonlinearities.

Contribution

It introduces a detailed Morse index computation for radial solutions and proves bifurcation of nonradial solutions using bifurcation theory.

Findings

Morse index of radial solutions is explicitly computed.

Existence of nonradial solutions bifurcates from radial ones.

Special case with exponential nonlinearity yields detailed insights.

Abstract

In this paper we study the problem -\Delta u =\left(\frac{2+\alpha}{2}\right)^2\abs{x}^{\alpha}f(\lambda,u), & \hbox{in}B_1 \\ u > 0, & \hbox{in}B_1 u = 0, & \hbox{on} \partial B_1 where $B_1$ is the unit ball of $\R^2$, $f$ is a smooth nonlinearity and $\a$, $\l$ are real numbers with $\a>0$. From a careful study of the linearized operator we compute the Morse index of some radial solutions to \eqref{i0}. Moreover, using the bifurcation theory, we prove the existence of branches of nonradial solutions for suitable values of the positive parameter $\l$. The case $f(\lambda,u)=\l e^u$ provides more detailed information.