## Topological Methods in Nonlinear Analysis

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

Giuseppina Vannella

#### Abstract

The following $T^2$-equivariant problem of periodic type is considered: $$\begin{cases} u\in C^2({\mathbb R}^2,{\mathbb R}),\\ -\varepsilon\Delta u(x,y)+F'(u(x,y))=0 & \text{in {\mathbb R}^{2},}\\ u(x,y)=u(x+T,y)=u(x,y+S) &\text{for all (x,y)\in {\mathbb R}^2,}\\ \nabla u(x,y)=\nabla u(x+T,y)=\nabla u(x,y+S) &\text{for all (x,y)\in {\mathbb R}^{2}.} \end{cases}\tag{\text{P}}$$ 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 $T^2$.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 17, Number 1 (2001), 41-53.

Dates
First available in Project Euclid: 22 August 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1471875796

Mathematical Reviews number (MathSciNet)
MR1846977

Zentralblatt MATH identifier
0992.35035

#### Citation

Vannella, Giuseppina. Morse theory applied to a $T^{2}$-equivariant problem. Topol. Methods Nonlinear Anal. 17 (2001), no. 1, 41--53. https://projecteuclid.org/euclid.tmna/1471875796

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