### 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 $T^2$.

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 $T^2$.

#### Keywords

PDE; critical points; Morse theory; group actions

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