On Multiple Mixed Interior and Boundary Peak Solutions for Some Singularly Perturbed Neumann Problems

2000 ◽  
Vol 52 (3) ◽  
pp. 522-538 ◽  
Author(s):  
Changfeng Gui ◽  
Juncheng Wei
Keyword(s):  
Mean Curvature ◽  
Variational Approach ◽  
Existence Of Solutions ◽  
Sphere Packing ◽  
Curvature Function ◽  
Bounded Smooth Domain ◽  
Neumann Problems ◽  
Subcritical Nonlinearity ◽  
The Mean ◽  
Bounded Smooth

AbstractWe consider the problemwhere Ω is a bounded smooth domain in RN, ε > 0 is a small parameter and f is a superlinear, subcritical nonlinearity. It is known that this equation possesses multiple boundary spike solutions that concentrate, as ε approaches zero, at multiple critical points of the mean curvature function H(P), P ∈ ∂Ω. It is also proved that this equation has multiple interior spike solutions which concentrate, as ε → 0, at sphere packing points in Ω.In this paper, we prove the existence of solutions with multiple spikes both on the boundary and in the interior. The main difficulty lies in the fact that the boundary spikes and the interior spikes usually have different scales of error estimation. We have to choose a special set of boundary spikes to match the scale of the interior spikes in a variational approach.

scholarly journals Existence of Solutions to Elliptic Problem with Convection Term and Lower-Order Term

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Xiaohua He ◽  
Shuibo Huang ◽  
Qiaoyu Tian ◽  
Yonglin Xu
Keyword(s):  
Dirichlet Problem ◽  
Lower Order ◽  
Elliptic Equations ◽  
Existence Of Solutions ◽  
Order Term ◽  
Combined Effects ◽  
Convection Term ◽  
Bounded Smooth Domain ◽  
Bounded Smooth ◽  
Lower Order Terms

In this paper, we establish the existence of solutions to the following noncoercivity Dirichlet problem − div M x ∇ u + u p − 1 u = − div u E x + f x , x ∈ Ω , u x = 0 , x ∈ ∂ Ω , where Ω ⊂ ℝ N N > 2 is a bounded smooth domain with 0 ∈ Ω , f belongs to the Lebesgue space L m Ω with m ≥ 1 , p > 0 . The main innovation point of this paper is the combined effects of the convection terms and lower-order terms in elliptic equations.

scholarly journals Existence of Solutions for a Modified Nonlinear Schrödinger System

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Yujuan Jiao ◽  
Yanli Wang
Keyword(s):  
Perturbation Method ◽  
Existence Of Solutions ◽  
Critical Sobolev Exponent ◽  
Smooth Domain ◽  
Bounded Smooth Domain ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrödinger System ◽  
Bounded Smooth ◽  
Sobolev Exponent ◽  
Schrödinger System

We are concerned with the following modified nonlinear Schrödinger system:-Δu+u-(1/2)uΔ(u2)=(2α/(α+β))|u|α-2|v|βu,  x∈Ω,  -Δv+v-(1/2)vΔ(v2)=(2β/(α+β))|u|α|v|β-2v,  x∈Ω,  u=0,  v=0,  x∈∂Ω, whereα>2,  β>2,  α+β<2·2*,  2*=2N/(N-2)is the critical Sobolev exponent, andΩ⊂ℝN  (N≥3)is a bounded smooth domain. By using the perturbation method, we establish the existence of both positive and negative solutions for this system.

On the interior spike solutions for some singular perturbation problems

1998 ◽  
Vol 128 (4) ◽  
pp. 849-874 ◽  
Author(s):  
Juncheng Wei
Keyword(s):  
Weak Convergence ◽  
Singular Perturbation ◽  
Viscosity Solutions ◽  
Sufficient Conditions ◽  
Bounded Smooth Domain ◽  
Neumann Problems ◽  
Weak Convergence Of Measures ◽  
Bounded Smooth ◽  
Necessary And Sufficient ◽  
Perturbation Problems

For some singular perturbed Dirichlet and Neumann problems in a bounded smooth domain, we study solutions which have a spike in the interior. We obtain both necessary and sufficient conditions for the existence of interior spike solutions. We use, among others, the methods of projections and viscosity solutions, weak convergence of measures and Liapunov–Schmidt reduction.

scholarly journals On the mean curvature function for compact surfaces

1981 ◽  
Vol 16 (2) ◽  
pp. 179-183 ◽  
Author(s):  
H. Blaine Lawson, Jr. ◽  
Renato de Azevedo Tribuzy
Keyword(s):  
Mean Curvature ◽  
Curvature Function ◽  
The Mean ◽  
Compact Surfaces

ON THE MEAN CURVATURE FUNCTION FOR COMPACT SURFACES

1983 ◽  
pp. 141-145
Author(s):  
H. BLAINE LAWSON ◽  
RENATO DE AZEVEDO TRIBUZY
Keyword(s):  
Mean Curvature ◽  
Curvature Function ◽  
The Mean ◽  
Compact Surfaces

scholarly journals THE MEAN CURVATURE EQUATION ON SEMIDIRECT PRODUCTS : HEIGHT ESTIMATES AND SCHERK-LIKE GRAPHS

2016 ◽  
Vol 101 (1) ◽  
pp. 118-144
Author(s):  
ÁLVARO KRÜGER RAMOS
Keyword(s):  
Mean Curvature ◽  
Smooth Curve ◽  
Quasilinear Elliptic Equations ◽  
Curvature Function ◽  
Minimal Graph ◽  
Semidirect Products ◽  
Height Estimates ◽  
Mean Curvature Equation ◽  
The Mean ◽  
Quasilinear Elliptic

In the ambient space of a semidirect product $\mathbb{R}^{2}\rtimes _{A}\mathbb{R}$, we consider a connected domain ${\rm\Omega}\subseteq \mathbb{R}^{2}\rtimes _{A}\{0\}$. Given a function $u:{\rm\Omega}\rightarrow \mathbb{R}$, its ${\it\pi}$-graph is $\text{graph}(u)=\{(x,y,u(x,y))\mid (x,y,0)\in {\rm\Omega}\}$. In this paper we study the partial differential equation that $u$ must satisfy so that $\text{graph}(u)$ has prescribed mean curvature $H$. Using techniques from quasilinear elliptic equations we prove that if a ${\it\pi}$-graph has a nonnegative mean curvature function, then it satisfies some uniform height estimates that depend on ${\rm\Omega}$ and on the supremum the function attains on the boundary of ${\rm\Omega}$. When $\text{trace}(A)>0$, we prove that the oscillation of a minimal graph, assuming the same constant value $n$ along the boundary, tends to zero when $n\rightarrow +\infty$ and goes to $+\infty$ if $n\rightarrow -\infty$. Furthermore, we use these estimates, allied with techniques from Killing graphs, to prove the existence of minimal ${\it\pi}$-graphs assuming the value zero along a piecewise smooth curve ${\it\gamma}$ with endpoints $p_{1},\,p_{2}$ and having as boundary ${\it\gamma}\cup (\{p_{1}\}\times [0,\,+\infty ))\cup (\{p_{2}\}\times [0,\,+\infty ))$.

VARIATIONAL APPROACH TO THE PHASE DIAGRAM OF RIGID MEMBRANES

1993 ◽  
Vol 07 (27) ◽  
pp. 4615-4629
Author(s):  
U. MARINI BETTOLO MARCONI ◽  
A. MARITAN
Keyword(s):  
Phase Transition ◽  
Phase Diagram ◽  
Euclidean Space ◽  
Mean Curvature ◽  
Variational Approach ◽  
Gaussian Approximation ◽  
Dimensional Euclidean Space ◽  
Bending Rigidity ◽  
Elastic Networks ◽  
The Mean

D-dimensional elastic networks randomly embedded in a d>D dimensional euclidean space, are studied employing Hartree (Gaussian) approximation. In presence of an energy depending on the mean curvature this approach leads to the prediction of a phase transition between a flat and a crumpled regime as the bending rigidity decreases in agreement with previous approximate calculations.

New solutions for critical Neumann problems in ℝ2

2017 ◽  
Vol 8 (1) ◽  
pp. 615-644
Author(s):  
Shengbing Deng ◽  
Monica Musso
Keyword(s):  
Boundary Condition ◽  
Elliptic Equation ◽  
Small Parameter ◽  
Neumann Boundary Condition ◽  
Neumann Boundary ◽  
Normal Vector ◽  
Bounded Smooth Domain ◽  
Neumann Problems ◽  
Bounded Smooth ◽  
Outer Normal Vector

Abstract We consider the elliptic equation {-\Delta u+u=0} in a bounded, smooth domain Ω in {\mathbb{R}^{2}} subject to the nonlinear Neumann boundary condition {\frac{\partial u}{\partial\nu}=\lambda ue^{u^{2}}} , where ν denotes the outer normal vector of {\partial\Omega} . Here {\lambda>0} is a small parameter. For any λ small we construct positive solutions concentrating, as {\lambda\to 0} , around points of the boundary of Ω.

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