Symmetry breaking for sign changing solutions of some elliptic problems

Abstract:

In the first part of this talk, we consider the nonlinear Hénon equation on a ball and show that, as a parameter goes to infinity, nonradial solutions bifurcate from (nodal) radial solutions. Our main tool is a thorough analysis of the eigenvalues of associated linearized operators which is based on a suitable rescaling of the equation. This allows us to identify a limit problem and yields asymptotic estimates for these eigenvalues.

The second part is concerned with solutions of a nonlinear Schrödinger equation which are invariant with respect to a spiraling motion. This leads to the study of least energy sign-changing solutions for an associated elliptic equation on the plane, for which we show symmetry breaking as the rotational slope increases.

This is based on joint works with Tobias Weth and Oscar Agudelo.