Nonnegative solutions with a nontrivial nodal set for elliptic equations on smooth symmetric domains

TL;DR

This paper constructs examples of nonnegative solutions to symmetric elliptic equations on smooth domains that have nontrivial nodal sets, challenging the typical symmetry and monotonicity properties known for positive solutions.

Contribution

It provides the first known examples of smooth symmetric domains where nonnegative solutions exhibit non-monotonic behavior, with nontrivial nodal sets, extending prior nonsmooth domain results.

Findings

Examples of solutions with nontrivial nodal sets on smooth domains.

Demonstration that nonnegative solutions can lack symmetry in monotonicity.

Extension of known phenomena from nonsmooth to smooth domains.

Abstract

We consider a semilinear elliptic equation on a smooth bounded domain $\Om$ in $\R^2$, assuming that both the domain and the equation are invariant under reflections about one of the coordinate axes, say the y-axis. It is known that nonnegative solutions of the Dirichlet problem for such equations are symmetric about the axis, and, if strictly positive, they are also decreasing in $x$ for $x>0$. Our goal is to exhibit examples of equations which admit nonnegative, nonzero solutions for which the second property fails; necessarily, such solutions have a nontrivial nodal set in $\Om$. Previously, such examples were known for nonsmooth domains only.