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

Authors: P. Poláčik and Susanna Terracini
Journal: Proc. Amer. Math. Soc. 142 (2014), 1249-1259
MSC (2010): Primary 35J61, 35B06, 35B05
Published electronically: January 8, 2014
MathSciNet review: 3162247
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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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Additional Information

P. Poláčik
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

Susanna Terracini
Affiliation: Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Piazza Ateneo Nuovo 1, 20126 Milano, Italy

Keywords: Semilinear elliptic equation, planar domain, nonnegative solutions, nodal set
Received by editor(s): April 30, 2012
Published electronically: January 8, 2014
Additional Notes: The first author was supported in part by NSF grant DMS-0900947
The second author was supported in part by PRIN2009 grant “Critical Point Theory and Perturbative Methods for Nonlinear Differential Equations”
Communicated by: Yingfei Yi
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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