TY - JOUR
T1 - Nonnegative solutions with a nontrivial nodal set for elliptic equations on smooth symmetric domains
AU - Poláčik, P.
AU - Terracini, Susanna
PY - 2014
Y1 - 2014
N2 - We consider a semilinear elliptic equation on a smooth bounded domain Ω in ℝ2, 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 Ω. Previously, such examples were known for nonsmooth domains only.
AB - We consider a semilinear elliptic equation on a smooth bounded domain Ω in ℝ2, 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 Ω. Previously, such examples were known for nonsmooth domains only.
KW - Nodal set
KW - Nonnegative solutions
KW - Planar domain
KW - Semilinear elliptic equation
UR - http://www.scopus.com/inward/record.url?scp=84893306423&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84893306423&partnerID=8YFLogxK
U2 - 10.1090/S0002-9939-2014-11942-3
DO - 10.1090/S0002-9939-2014-11942-3
M3 - Article
AN - SCOPUS:84893306423
SN - 0002-9939
VL - 142
SP - 1249
EP - 1259
JO - Proceedings of the American Mathematical Society
JF - Proceedings of the American Mathematical Society
IS - 4
ER -