Solutions on a torus for a semilinear equation

  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},
  pages={371 - 382}
  • G. Allain, Anne Beaulieu
  • Published 1 April 2011
  • Mathematics
  • Proceedings of the Royal Society of Edinburgh: Section A Mathematics
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. 


New solutions of equations on $\mathbb {R}^n$
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
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
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
It is proved that exactly the same estimate is true when the period is 2 pi/epsilon, even when epsilon tends to 0.
On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type
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
The One-Dimensional Maximum Principle.- Elliptic Equations.- Parabolic Equations.- Hyperbolic Equations.- Bibliography.- Index.
Uniqueness of positive solutions of Δu−u+up=0 in Rn
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
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