The following $T^2$-equivariant problem of periodic type is considered: $$\cases u\in C^2({\mathbb R}^2,{\mathbb R}),\cr -\varepsilon\Delta u(x,y)+F^{\prime }(u(x,y))=0 & \text{in {\mathbb R}^{2},}\cr u(x,y)=u(x+T,y)=u(x,y+S) &\text{for all (x,y)\in {\mathbb R}^2,}\cr \nabla u(x,y)=\nabla u(x+T,y)=\nabla u(x,y+S) &\text{for all (x,y)\in {\mathbb R}^{2}.} \endcases\tag{\text{P}}$$ Using a suitable version of Morse theory for equivariant problems, it is proved that an arbitrarily great number of orbits of solutions to (P) is founded, choosing $\varepsilon> 0$ suitably small. Each orbit is homeomorphic to $S^1$ or to $T^2$.

### Morse theory applied to a $T^{2}$-equivariant problem

#### Abstract

The following $T^2$-equivariant problem of periodic type is considered: $$\cases u\in C^2({\mathbb R}^2,{\mathbb R}),\cr -\varepsilon\Delta u(x,y)+F^{\prime }(u(x,y))=0 & \text{in {\mathbb R}^{2},}\cr u(x,y)=u(x+T,y)=u(x,y+S) &\text{for all (x,y)\in {\mathbb R}^2,}\cr \nabla u(x,y)=\nabla u(x+T,y)=\nabla u(x,y+S) &\text{for all (x,y)\in {\mathbb R}^{2}.} \endcases\tag{\text{P}}$$ Using a suitable version of Morse theory for equivariant problems, it is proved that an arbitrarily great number of orbits of solutions to (P) is founded, choosing $\varepsilon> 0$ suitably small. Each orbit is homeomorphic to $S^1$ or to $T^2$.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11589/11619
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