Nonnegative solutions with a nontrivial nodal set for elliptic equations on smooth symmetric domains
نویسندگان
چکیده
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.
منابع مشابه
A discussion of nonnegative solutions of elliptic equations on symmetric domains∗
In this note we summarize our recent results on nonnegative solutions of nonlinear elliptic equations on reflectionally symmetric domains. We discuss symmetry properties of such solutions, the structure of their nodal set, and the existence and multiplicity of solutions with a nontrivial nodal set.
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تاریخ انتشار 2012