Morse theory applied to a T^2-equivariant problem - Vannella, Giuseppina (Topological Methods in Nonlinear Analysis, 2001)
The following T^2-equivariant problem of periodic type is considered: uC^2(R^2,R),-εΔ u(x,y)+F^′ (u(x,y))=0 & in R^2,u(x,y)=u(x+T,y)=u(x,y+S) &for all (x,y)R^2,abla u(x,y)= abla u(x+T,y)= abla u(x,y+S) &for all (x,y)R^2. P Using a suitable version of Morse theory for equivariant problems, it is proved that an arbitrarily great number of orbits of solutions to (P) is founded, choosing ε> 0 suitably small. Each orbit is homeomorphic to S^1 or to T^2.