Infinitely many radially symmetric solutions of certain semilinear elliptic equations (Q1191888)

For all \(k \in \mathbb{N}\) the existence of solutions \(u(x)=\nu \bigl( | x | \bigr)\) to \(\Delta u+f(u)=0\) in \(\mathbb{R}^ n\), \(u(\infty)=0\), with \(\nu\) having exactly \(k\) zeros, is shown, provided that \(f\) satisfies \(sf(s)>0\), \(s \neq 0\) and further conditions. It is emphasized that the literature covers mainly the reverse case \(sf(s)<0\). For \(n=2\) a statement concerning the existence of infinitely many solutions is made.