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A counterexample to the nodal domain conjecture and a related semilinear equation. (English) Zbl 0652.35085
The authors study the eigenvalue problem $-(\Delta +V)\phi +\lambda \phi =0\quad in\quad \Omega;\quad \phi =0\quad on\quad \partial \Omega,$ where V is a given smooth function, $$\lambda$$ is an eigenvalue and $$\Omega$$ is a bounded smooth domain in $$R^ n$$, $$n\geq 2$$. The set $$\{$$ $$x\in \Omega |$$ $$\phi (x)=0\}$$ is called the nodal set of $$\phi$$. The following conjecture has been around for quite some time.
Nodal Domain Conjecture: The nodal line of any second eigenfunction must intersect the boundary $$\partial \Omega$$ at exactly two points. In the paper the authors construct a counterexample to show that the above conjecture is false when $$V\not\equiv 0$$.
Reviewer: C.F.Wang

##### MSC:
 35P05 General topics in linear spectral theory for PDEs 35J25 Boundary value problems for second-order elliptic equations 35J60 Nonlinear elliptic equations
##### Keywords:
eigenvalue problem; Nodal Domain Conjecture
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##### References:
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