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

Poláčik, P.; Terracini, Susanna

doi:10.1090/S0002-9939-2014-11942-3

Nonnegative solutions with a nontrivial nodal set for elliptic equations on smooth symmetric domains
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by P. Poláčik and Susanna Terracini PDF

Proc. Amer. Math. Soc. 142 (2014), 1249-1259 Request permission

Abstract:

We consider a semilinear elliptic equation on a smooth bounded domain $\Omega$ in $\mathbb {R}^2$, 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 $\Omega$. Previously, such examples were known for nonsmooth domains only.

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