HARDY-SOBOLEV INEQUALITY IN H1(Ω) AND ITS APPLICATIONS

2002 ◽  
Vol 04 (03) ◽  
pp. 409-434 ◽  
Author(s):  
ADIMURTHI
Keyword(s):  
Boundary Condition ◽  
Sobolev Inequality ◽  
Neumann Boundary Condition ◽  
Remainder Term ◽  
Blow Up ◽  
Neumann Boundary ◽  
Smooth Domain ◽  
Bounded Smooth Domain ◽  
Eigen Value ◽  
Bounded Smooth

In this article, we have determined the remainder term for Hardy–Sobolev inequality in H1(Ω) for Ω a bounded smooth domain and studied the existence, non existence and blow up of first eigen value and eigen function for the corresponding Hardy–Sobolev operator with Neumann boundary condition.

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.

scholarly journals Blow-Up Analysis for a Quasilinear Parabolic Equation with Inner Absorption and Nonlinear Neumann Boundary Condition

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Zhong Bo Fang ◽  
Yan Chai
Keyword(s):  
Boundary Condition ◽  
Parabolic Equation ◽  
Neumann Boundary Condition ◽  
Blow Up ◽  
Quasilinear Parabolic Equation ◽  
Neumann Boundary ◽  
Initial Boundary ◽  
Blow Up Analysis ◽  
Inner Absorption ◽  
Quasilinear Parabolic

We investigate an initial-boundary value problem for a quasilinear parabolic equation with inner absorption and nonlinear Neumann boundary condition. We establish, respectively, the conditions on nonlinearity to guarantee thatu(x,t)exists globally or blows up at some finite timet*. Moreover, an upper bound fort*is derived. Under somewhat more restrictive conditions, a lower bound fort*is also obtained.

Further study of a fourth-order elliptic equation with negative exponent

2011 ◽  
Vol 141 (3) ◽  
pp. 537-549 ◽  
Author(s):  
Zongming Guo ◽  
Zhongyuan Liu
Keyword(s):  
Fourth Order ◽  
Blow Up ◽  
Smooth Domain ◽  
Regular Solutions ◽  
Order Problem ◽  
Bounded Smooth Domain ◽  
Main Technique ◽  
Fourth Order Elliptic Equation ◽  
Fourth Order Problem ◽  
Bounded Smooth

We continue to study the nonlinear fourth-order problem TΔu – DΔ2u = λ/(L + u)2, –L < u < 0 in Ω, u = 0, Δu = 0 on ∂Ω, where Ω ⊂ ℝN is a bounded smooth domain and λ > 0 is a parameter. When N = 2 and Ω is a convex domain, we know that there is λc > 0 such that for λ ∊ (0, λc) the problem possesses at least two regular solutions. We will see that the convexity assumption on Ω can be removed, i.e. the main results are still true for a general bounded smooth domain Ω. The main technique in the proofs of this paper is the blow-up argument, and the main difficulty is the analysis of touch-down behaviour.

Optimal global asymptotic behaviour of the solution to a class of singular Dirichlet problems

2020 ◽  
pp. 1-19
Author(s):  
Zhijun Zhang
Keyword(s):  
Asymptotic Behaviour ◽  
Unique Solution ◽  
Blow Up ◽  
Critical Values ◽  
Smooth Domain ◽  
Dirichlet Problems ◽  
Bounded Smooth Domain ◽  
Bounded Smooth

This paper is mainly concerned with the global asymptotic behaviour of the unique solution to a class of singular Dirichlet problems − Δu = b(x)g(u), u > 0, x ∈ Ω, u|∂Ω = 0, where Ω is a bounded smooth domain in ℝ n , g ∈ C1(0, ∞) is positive and decreasing in (0, ∞) with $\lim _{s\rightarrow 0^+}g(s)=\infty$ , b ∈ Cα(Ω) for some α ∈ (0, 1), which is positive in Ω, but may vanish or blow up on the boundary properly. Moreover, we reveal the asymptotic behaviour of such a solution when the parameters on b tend to the corresponding critical values.

Structure of positive boundary blow-up solutions

2004 ◽  
Vol 134 (5) ◽  
pp. 923-938 ◽  
Author(s):  
Zongming Guo
Keyword(s):  
Boundary Layer ◽  
Blow Up ◽  
Smooth Domain ◽  
Bounded Smooth Domain ◽  
Semilinear Problems ◽  
Bounded Smooth ◽  
Positive Boundary

The structure of positive boundary blow-up solutions to semilinear problems of the form −Δu = λf(u) in Ω, u = ∞ on ∂Ω, Ω ⊂ RN a bounded smooth domain, is studied for a class of nonlinearities f ∈ C1 ([0, ∞)\{z2}) satisfying f (0) = f(z1) = f (z2) = 0 with 0 < z1 < z2, f < 0 in (0, z1)∪(z2, ∞), f > 0 in (z1, z2). Two positive boundary-layer solutions and infinitely many positive spike-layer solutions are obtained for λ sufficiently large.

BLOW-UP ANALYSIS FOR SOLUTIONS OF THE LAPLACIAN EQUATION WITH EXPONENTIAL NEUMANN BOUNDARY CONDITION IN DIMENSION TWO

2006 ◽  
Vol 08 (06) ◽  
pp. 737-761 ◽  
Author(s):  
YU-XIA GUO ◽  
JIA-QUAN LIU
Keyword(s):  
Boundary Condition ◽  
Asymptotic Behavior ◽  
Existence Theorem ◽  
Neumann Boundary Condition ◽  
Blow Up ◽  
Neumann Boundary ◽  
Asymptotic Behavior Of Solutions ◽  
Laplacian Equation ◽  
Blow Up Analysis

We consider the asymptotic behavior of solutions of the Laplacian equation with exponential Neumann boundary condition in dimension two. As an application, we prove an existence theorem of nonminimum solutions.

scholarly journals Weighted Steklov problem under nonresonance conditions

2018 ◽  
Vol 36 (4) ◽  
pp. 87-105
Author(s):  
Jonas Doumatè ◽  
Aboubacar Marcos
Keyword(s):  
Boundary Condition ◽  
Nonlinear Problem ◽  
Weak Solutions ◽  
Existence Results ◽  
Smooth Domain ◽  
Existence Of Weak Solutions ◽  
Steklov Problem ◽  
Bounded Smooth Domain ◽  
Carathéodory Function ◽  
Bounded Smooth

We deal with the existence of weak solutions of the nonlinear problem $-\Delta_{p}u+V|u|^{p-2}u$ in a bounded smooth domain $\Omega\subset \mathbb{R}^{N}$ which is subject to the boundary condition $|\nabla u|^{p-2}\frac{\partial u}{\partial \nu}=f(x,u)$. Here $V\in L^{\infty}(\Omega)$ possibly exhibit both signs which leads to an extension  of particular cases in literature and $f$ is a Carathéodory function that satisfies some additional conditions. Finally we prove, under and between nonresonance condtions, existence results for the problem.

Sign in / Sign up

Export Citation Format

Share Document