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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
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