Necessary and sufficient condition for existence and uniqueness of nodal solutions to sublinear elliptic equations (Q5954462)

Let \(B\) be an arbitrary ball in \(\mathbb{R}^n\) and let \(f:\mathbb{R}\rightarrow \mathbb{R}\) be a continuous function with sublinear growth at infinity. Some standard nonlinearities satisfying the assumptions imposed in the paper are: (i) \(f(u)=|u|^p\text{ sgn} u\), for \(0<p<1\); (ii) \(f(u)=|u|^p\text{ sgn} u-u\), for \(0<p<1\) ; (iii) \(f(u)=-u\log |u|\). The paper deals with the semilinear elliptic equation \(\Delta u+f(u)=0\) in \(B\), subject to the Dirichlet boundary condition \(u=0\) on \(\partial B\). The main result of the paper establishes a necessary and sufficient condition for the existence and uniqueness of radially symmetric nodal solutions to the above nonlinear problem.