科研之友
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摘要
We consider a semilinear elliptic equation on a smooth bounded domain Omega in 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.
- 出版日期2014-4
