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.References
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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
- 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
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 142 (2014), 1249-1259
- MSC (2010): Primary 35J61, 35B06, 35B05
- DOI: https://doi.org/10.1090/S0002-9939-2014-11942-3
- MathSciNet review: 3162247