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 typeis considered:$$\cases u\in C^2({\mathbb R}^2,{\mathbb R}),\cr-εΔ u(x,y)+F^{′ }(u(x,y))=0 & \text{in ${\mathbb R}^{2}$,}\cru(x,y)=u(x+T,y)=u(x,y+S) &\text{for all $(x,y)\in {\mathbb R}^2$,}\cr abla u(x,y)= abla u(x+T,y)= abla u(x,y+S) &\text{for all $(x,y)\in {\mathbb R}^{2}$.}\endcases\tag{\text{P}}$$Using a suitable version of Morse theory for equivariantproblems, it is proved that an arbitrarily great number of orbitsof solutions to (P) is founded, choosing $ε> 0$suitably small. Each orbit is homeomorphic to $S^1$ or to $T^2$.