Radially symmetric solutions of semilinear elliptic equations, existence and Sobolev estimates (Q1174681)

The semilinear elliptic equation \(\Delta u+f(| x|,u)=0\) in the unit ball of \(\mathbb{R}^ n\) is converted into the ordinary differential equation \(u''+((n-1)/t)u'+f(t,u)=0.\) The author assumes the boundary conditions \(u'(0)=0\), \(au(1)+bu'(1)=0\) for any coefficients \(a\) and \(b\), and proves four theorems. The first two assert that there exist infinitely many solutions when \(f\) is assumed to have superlinear or sublinear growth. In the last two \(H^ 1\) estimates are given in terms of the number of the zeros of the solutions. The proofs are combinations of the shooting method, a priori estimates for solutions, and new types of Prüfer transformations.