# Solutions on a torus for a semilinear equation

@article{Allain2011SolutionsOA, title={Solutions on a torus for a semilinear equation}, author={Genevi{\`e}ve Allain and Anne Beaulieu}, journal={Proceedings of the Royal Society of Edinburgh: Section A Mathematics}, year={2011}, volume={141}, pages={371 - 382} }

We are interested in the positive doubly periodic solutions, which are even in each variable, of a stationary nonlinear Schrödinger equation in ℝ2, with a small parameter. For any pair of periods (2a, 2b), we construct a branch of solutions that concentrate uniformly to the ground-state solution of the equation.

## References

SHOWING 1-10 OF 15 REFERENCES

New solutions of equations on $\mathbb {R}^n$

- Mathematics
- 2001

We consider some weakly nonlinear elliptic equations on the whole
space and use local and global bifurcations methods to construct solutions periodic
in one variable and decaying in the other…

A new type of concentration solutions for a singularly perturbed elliptic problem

- Mathematics
- 2006

We prove the existence of positive solutions concentrating on some higher dimensional manifolds near the boundary of the domain for a nonlinear singularly perturbed elliptic problem.

Semilinear Neumann boundary value problems on a rectangle

- Mathematics
- 2002

We consider a semilinear elliptic equation Δu + λf(u) = 0, x ∈ Ω, ∂u/∂n = 0, x ∈ ∂Ω, where Ω is a rectangle (0, a) x (0, b) in R 2 . For balanced and unbalanced f, we obtain partial descriptions of…

An a priori estimate for the singly periodic solutions of a semilinear equation

- MathematicsAsymptot. Anal.
- 2012

It is proved that exactly the same estimate is true when the period is 2 pi/epsilon, even when epsilon tends to 0.

Existence and instability of spike layer solutions to singular perturbation problems

- Mathematics
- 2002

On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type

- Mathematics
- 1986

On deduit des estimations a priori pour des solutions positives du probleme de Neumann pour des systemes elliptiques semilineaires ainsi que pour des equations isolees semilineaires reliees a ces…

Maximum principles in differential equations

- Mathematics
- 1967

The One-Dimensional Maximum Principle.- Elliptic Equations.- Parabolic Equations.- Hyperbolic Equations.- Bibliography.- Index.

Uniqueness of positive solutions of Δu−u+up=0 in Rn

- Mathematics
- 1989

We establish the uniqueness of the positive, radially symmetric solution to the differential equation Δu−u+up=0 (with p>1) in a bounded or unbounded annular region in Rn for all n≧1, with the Neumann…

Elliptic Partial Differential Equations of Second Order

- Mathematics
- 1997

We study in this chapter a class of partial differential equations that generalize and are to a large extent represented by Laplace’s equation. These are the elliptic partial differential equations…