Infinitely many radial solutions of an elliptic system
The system \[ \Delta u_ 1+f_ 1(u_ 1,u_ 2)=0,\quad \Delta u_ 2+f_ 2(u_ 1,u_ 2)=0,\quad u_ i=u_ i(x),\quad x\in {\mathbb{R}}\quad n,\quad n>1, \] is studied with \(f_ i=\partial F/\partial u_ i\) for some \(F\in C^ 2({\mathbb{R}}^ 2)\). Under some technical assumptions on F two main results are presented: The existence of infinitely many radial solutions and the existence of a radial solution with prescribed winding number K or \(K+1\), \(K\in {\mathbb{N}}\). The proof is split up into three papers of which two are forthcoming. In the paper in question the notion of a winding number - counting how many times a trajectory winds around certain two dimensional phases - is introduced. A family of equations depending on \(\epsilon >0\) is studied and the existence of infinitely many solutions to this perturbed problem is shown. A geometrical proof is given. As \(\epsilon\to 0\) some sequence of the set of solutions with winding number K (assumed here to be given) will converge to a solution of the original problem with winding number K. For this, a priori estimates are proved.
scientific article; zbMATH DE number 4063627