Nonnegative solutions with a nontrivial nodal set for elliptic equations on smooth symmetric domains
Peter Poláčik, Susanna Terracini · Proceedings of the American Mathematical Society · 2014
We consider a semilinear elliptic equation on a smooth bounded domain Ω \Omega in R 2 \mathbb {R}^2 , assuming that both the domain and the equation are invariant under reflections about one of the coordinate axes, say the y 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 x for x > 0 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 Ω \Omega . Previously, such examples were known for nonsmooth domains only.