A note on the radial symmetry of solutions with spherical nodal sets (Q1877819)

The author studies solutions of \(u(x)\) of (1) \( \Delta=f(u)\), (2) \(u|_{\mathbb B_R}=0\) where \(u\in C^2 (\mathbb B_R)\), \(\mathbb B_R\) is an \(N\)-dimensional ball with radius \(R,\) \(N\geq 3\). In the present study the author is mainly interested in the solutions \(u(x )\) of (1)--(2) having the property: the nodal set of \(u(x)\) is spherical and \(u(0)>0\). The author investigates its radial symmetrical property.