A positive solution of a nonlinear elliptic equation in \(\mathbb R^N\) with \(G\)-symmetry. (English) Zbl 1166.35015

The equation studied in the paper is the nonlinear differential equation \(-\Delta u+u=f(x,u)\) for functions \(u\in H^1(\mathbb R^n)\). Here the nonlinearity \(f(x,u)\) is supposed to be invariant in the first variable under the action of a finite subgroup \(G\) of \(O(n)\). Under quite a number of assumptions on \(f\), depending also on \(G\) and its action on \(N\), the author proves existence of a positive \(G\)-invariant solution to the equation.
The proof uses the mountain pass theorem together with estimates for the \(G\)-symmetric mountain pass level which is shown to be strictly less than the first level on which the Palais-Smale condition may break down.


35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces