共查询到20条相似文献,搜索用时 31 毫秒
1.
Guantie Deng 《Bulletin des Sciences Mathématiques》2007,131(1):53
In this paper, using a modified Poisson kernel in an upper half-space, we prove that a harmonic function u(z) in a upper half space with its positive part u+(x)=max{u(x),0} satisfying a slowly growing condition can be represented by its integral in the boundary of the upper half space, the integral representation is unique up to the addition of a harmonic polynomial, vanishing in the boundary of the upper half space and that its negative part u−(x)=max{−u(x),0} can be dominated by a similar slowly growing condition, this improves some classical result about harmonic functions in the upper half space. 相似文献
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3.
G.A. Afrouzi 《Journal of Mathematical Analysis and Applications》2005,303(1):342-349
In this paper we shall study the following variant of the logistic equation with diffusion:
−du″(x)=g(x)u(x)−u2(x) 相似文献
4.
K.J. Brown 《Journal of Differential Equations》2003,193(2):481-499
The Nehari manifold for the equation −Δu(x)=λa(x)u(x)+b(x)|u(x)|ν−1u(x) for x∈Ω together with Dirichlet boundary conditions is investigated. Exploiting the relationship between the Nehari manifold and fibrering maps (i.e., maps of the form t→J(tu) where J is the Euler functional associated with the equation) we discuss how the Nehari manifold changes as λ changes and show how existence and non-existence results for positive solutions of the equation are linked to properties of the manifold. 相似文献
5.
Ricardo Enguiça 《Nonlinear Analysis: Theory, Methods & Applications》2010,73(9):2968-2979
We start by studying the existence of positive solutions for the differential equation
u″=a(x)u−g(u), 相似文献
6.
Özkan Öcalan 《Journal of Mathematical Analysis and Applications》2007,331(1):644-654
In this paper, we provide oscillation properties of every solution of the neutral differential equation with positive and negative coefficients
[x(t)−R(t)x(t−r)]′+P(t)x(t−τ)−Q(t)x(t−σ)=0, 相似文献
7.
Miguel Ramos 《Journal of Mathematical Analysis and Applications》2009,352(1):246-258
We study the existence, multiplicity and shape of positive solutions of the system −ε2Δu+V(x)u=K(x)g(v), −ε2Δv+V(x)v=H(x)f(u) in RN, as ε→0. The functions f and g are power-like nonlinearities with superlinear and subcritical growth at infinity, and V, H, K are positive and locally Hölder continuous. 相似文献
8.
Let u? be a single layered radially symmetric unstable solution of the Allen-Cahn equation −?2Δu=u(u−a(|x|))(1−u) over the unit ball with Neumann boundary conditions. We estimate the small eigenvalues of the linearized eigenvalue problem at u? when ? is small. As a consequence, we prove that the Morse index of u? is asymptotically given by [μ∗+o(1)]?−(N−1)/2 with μ∗ a certain positive constant expressed in terms of parameters determined by the Allen-Cahn equation. Our estimates on the small eigenvalues have many other applications. For example, they may be used in the search of other non-radially symmetric solutions, which will be considered in forthcoming papers. 相似文献
9.
We consider a dissipative version of the modified Korteweg-de Vries equation ut+uxxx−uxx+x(u3)=0. We prove global well-posedness results on the associated Cauchy problem in the Sobolev spaces Hs(R) for s>−1/4 while for s<−1/2 we prove some ill-posedness issues. 相似文献
10.
Songzhe Lian Chunling Cao Hongjun Yuan 《Journal of Mathematical Analysis and Applications》2008,342(1):27-38
The authors of this paper study the Dirichlet problem of the following equation
ut−div(|u|ν(x,t)∇u)=f−|u|p(x,t)−1u. 相似文献
11.
Francisco Odair de Paiva Eugenio Massa 《Journal of Mathematical Analysis and Applications》2008,342(1):638-650
We consider the Dirichlet problem for the equation −Δu=λu±f(x,u)+h(x) in a bounded domain, where f has a sublinear growth and h∈L2. We find suitable conditions on f and h in order to have at least two solutions for λ near to an eigenvalue of −Δ. A typical example to which our results apply is when f(x,u) behaves at infinity like a(x)|u|q−2u, with M>a(x)>δ>0, and 1<q<2. 相似文献
12.
Lucas C.F. Ferreira 《Journal of Differential Equations》2011,250(4):2045-2063
We study the equation Δu+u|u|p−1+V(x)u+f(x)=0 in Rn, where n?3 and p>n/(n−2). The forcing term f and the potential V can be singular at zero, change sign and decay polynomially at infinity. We can consider anisotropic potentials of form h(x)|x|−2 where h is not purely angular. We obtain solutions u which blow up at the origin and do not belong to any Lebesgue space Lr. Also, u is positive and radial, in case f and V are. Asymptotic stability properties of solutions, their behavior near the singularity, and decay are addressed. 相似文献
13.
Shingo Takeuchi 《Journal of Differential Equations》2011,251(8):2196-2208
This paper concerns the formation of a coincidence set for the positive solution of the boundary value problem: −εΔpu=uq−1f(a(x)−u) in Ω with u=0 on ∂Ω, where ε is a positive parameter, Δpu=div(|∇u|p−2∇u), 1<q?p<∞, f(s)∼|s|θ−1s(s→0) for some θ>0 and a(x) is a positive smooth function satisfying Δpa=0 in Ω with infΩ|∇a|>0. It is proved in this paper that if 0<θ<1 the coincidence set Oε={x∈Ω:uε(x)=a(x)} has a positive measure for small ε and converges to Ω with order O(ε1/p) as ε→0. Moreover, it is also shown that if θ?1, then Oε is empty for any ε>0. The proofs rely on comparison theorems and the energy method for obtaining local comparison functions. 相似文献
14.
Jörg Härterich 《Journal of Mathematical Analysis and Applications》2005,307(2):395-414
This paper deals with the singular limit for
L?u:=ut−Fx(u,?ux)−?−1g(u)=0, 相似文献
15.
Xiaojing Yang 《Journal of Differential Equations》2002,183(1):108-131
In this paper, the boundedness of all solutions of the nonlinear equation (?p(x′))′+(p-1)[α?p(x+)−β?p(x−)]+f(x)+g(x)=e(t) is discussed, where e(t)∈C7 is 2πp-periodic, f,g are bounded C6 functions, ?p(u)=∣u∣p−2u, p?2,α,β are positive constants, x+=max{x,0},x−=max{−x,0}. 相似文献
16.
We prove a sharp unique continuation theorem for nonnegative H2,1 solutions of the differential inequality |Δu(x)|?C−2|x−x0||u(x)| which vanish of finite order at x0. 相似文献
17.
G. Gripenberg 《Journal of Mathematical Analysis and Applications》2009,352(1):175-183
Sufficient conditions are given for the solutions to the (fully nonlinear, degenerate) elliptic equation F(x,u,Du,D2u)=0 in Ω to satisfy |u(x)−u(y)|?Cα|x−y| for some α∈(0,1) when x∈Ω and y∈∂Ω. 相似文献
18.
Jiangang Cheng 《Journal of Mathematical Analysis and Applications》2006,313(1):322-341
This paper is concerned with the exact number of positive solutions for the boundary value problem ′(|y′|p−2y′)+λf(y)=0 and y(−1)=y(1)=0, where p>1 and λ>0 is a positive parameter. We consider the case in which both f(u) and g(u)=(p−1)f(u)−uf′(u) change sign exactly once from negative to positive on (0,∞). 相似文献
19.
《Mathematical and Computer Modelling》1999,29(2):1-18
This paper is concerned with the construction of accurate continuous numerical solutions for partial self-adjoint differential systems of the type (P(t) ut)t = Q(t)uxx, u(0, t) = u(d, t) = 0, u(x, 0) = f(x), ut(x, 0) = g(x), 0 ≤ x ≤ d, t >- 0, where P(t), Q(t) are positive definite oRr×r-valued functions such that P′(t) and Q′(t) are simultaneously semidefinite (positive or negative) for all t ≥ 0. First, an exact theoretical series solution of the problem is obtained using a separation of variables technique. After appropriate truncation strategy and the numerical solution of certain matrix differential initial value problems the following question is addressed. Given T > 0 and an admissible error ϵ > 0 how to construct a continuous numerical solution whose error with respect to the exact series solution is smaller than ϵ, uniformly in D(T) = {(x, t); 0 ≤ x ≤ d, 0 ≤ t ≤ T}. Uniqueness of solutions is also studied. 相似文献
20.
Weilin Zou 《Nonlinear Analysis: Theory, Methods & Applications》2010,73(9):3069-3082
This paper deals with a class of degenerate quasilinear elliptic equations of the form −div(a(x,u,∇u)=g−div(f), where a(x,u,∇u) is allowed to be degenerate with the unknown u. We prove existence of bounded solutions under some hypothesis on f and g. Moreover we prove that there exists a renormalized solution in the case where g∈L1(Ω) and f∈(Lp′(Ω))N. 相似文献