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

Giuseppina Vannella

DOI: http://dx.doi.org/10.12775/TMNA.2001.003

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 $T^2$.

Keywords


PDE; critical points; Morse theory; group actions

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