# On the Boundary Spike Layer Solutions to a Singularly Perturbed Neumann Problem

@article{Wei1997OnTB, title={On the Boundary Spike Layer Solutions to a Singularly Perturbed Neumann Problem}, author={Juncheng Wei}, journal={Journal of Differential Equations}, year={1997}, volume={134}, pages={104-133} }

## 155 Citations

Dynamics of a boundary spike for the shadow Gierer-Meinhardt system

- Mathematics
- 2011

The Gierer-Meinhardt system is a mathematical
model describing the process of hydra regeneration.
The authors of [3] showed that
if an initial value is close to a spiky pattern and
its peak…

Locating the peaks of least-energy solutions to a quasilinear elliptic Neumann problem, Part II

- Mathematics
- 2010

Abstract We continue our work (Y. Li, C. Zhao, Locating the peaks of least-energy solutions to a quasilinear elliptic Neumann problem, J. Math. Anal. Appl. 336 (2007) 1368–1383) to study the shape of…

On the Multiplicity of Nodal Solutions to a Singularly Perturbed Neumann Problem

- Mathematics
- 2008

Abstract.We consider the problem
$$\varepsilon^{2}\Delta u + u = |u|^{p-1}\, u \,{\rm in} \, \Omega, \frac{\partial u}{\partial v}= 0\,{\rm on}\, \partial\Omega$$ where Ω is a bounded smooth domain…

Locating the Peaks of Least-Energy Solutions to a Quasilinear Elliptic Neumann Problem

- Mathematics
- 2007

Abstract In this paper we study the shape of least-energy solutions to the quasilinear problem e m Δ m u − u m − 1 + f ( u ) = 0 with homogeneous Neumann boundary condition. We use an intrinsic…

On the number of interior multipeak solutions for singularly perturbed Neumann problems

- Mathematics
- 1998

where e is a small positive number, Ω is a bounded domain in R with Cboundary, n is the unit outward normal of ∂Ω at y, 1 < p < (N + 2)/(N − 2) if N ≥ 3 and 1 < p <∞ if N = 2. Much work has been done…

Multiple boundary peak solutions for some singularly perturbed Neumann problems

- Mathematics
- 2000

Abstract We consider the problem
{ ɛ 2 Δ u - u + f ( u ) = 0 in Ω , u > 0 in Ω , ∂ u / ∂ v = 0 on ∂ Ω , where Ω is a bounded smooth domain in RN, ɛ > 0 is a small parameter and f is a superlinear,…

A P ] 2 2 A pr 2 02 1 Supercritical elliptic problems on nonradial annular domains via a non-smooth variational approach

- 2021

In this paper we are interested in positive classical solutions of −∆u = a(x)u in Ω, u > 0 in Ω, u = 0 on ∂Ω, (1) where Ω is a bounded annular domain (not necessarily an annulus) in R (N ≥ 3)…

On Ambrosetti-Malchiodi-Ni conjecture on two-dimensional smooth bounded domains: clustering concentration layers

- Mathematics
- 2021

We consider the clustering concentration on curves for solutions to the problem $$
\varepsilon^2 {\mathrm {div}}\big( \nabla_{{\mathfrak a}(y)} u\big)- V(y)u+u^p\, =\, 0, \quad u>0 \quad\mbox{in…

Supercritical elliptic problems on nonradial domains via a nonsmooth variational approach

- Mathematics
- 2021

In this paper we are interested in positive classical solutions of −∆u = a(x)u p−1 in Ω, u > 0 in Ω, u = 0 on ∂Ω, (1) where Ω is a bounded annular domain (not necessarily an annulus) in R (N ≥ 3)…

Boundary clustered layer positive solutions for an elliptic Neumann problem with large exponent.

- Physics
- 2020

Let $\mathcal{D}$ be a smooth bounded domain in $\mathbb{R}^N$ with $N\geq3$, we study the existence and profile of positive solutions for the following elliptic Neumann problem…

## References

SHOWING 1-10 OF 18 REFERENCES

On the Construction of Single-Peaked Solutions to a Singularly Perturbed Semilinear Dirichlet Problem

- Mathematics
- 1996

where 2= i=1 ( 2 xi ) is the Laplace operator, 0 is a bounded smooth domain in R, =>0 is a constant, and the exponent p satisfies 1< p< (n+2) (n&2) for n 3 and 1< p< for n=2. Problem (1.1) arises in…

Characterization of concentration points and L∞-estimates for solutions of a semilinear neumann problem involving the critical sobolev exponent

- Mathematics
- 1995

Let Ω ⊂ R n (n ≥ 7) be a bounded domain with smooth boundary. For λ > 0, let u λ be a solution of -Δu + λu= u n+2/n-2 in Ω. u > 0 in Ω, ∂u/∂v = 0 on ∂Ω, whose energy is less than the first critical…

Condensation of least-energy solutions of a semilinear Neumann problem

- J. Partial Differential Equations
- 1995

Further Study on the Effect of Boundary Conditions

- Mathematics
- 1995

Abstract This paper concerns the least-energy solutions to a semilinear elliptic equation. The role of geometry of the boundary is studied when a parameter in the boundary condition tends to a…

On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems

- Mathematics
- 1995

Point condensation generated by a reaction-diffusion system in axially symmetric domains

- Mathematics
- 1995

In this paper we consider the stationary problem for a reaction-diffusion system of activator-inhibitor type, which models biological pattern formation, in an axially symmetric domain. It is shown…

The role of mean curvature in a semilinear Neumann problem involving the critical Sobolev exponent, Comm

- Partial Differential Equations
- 1995