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1.
For n≥3 and p>1, the elliptic equation Δu+K(x)up+μf(x)=0 in possesses a continuum of positive entire solutions, provided that (i) locally Hölder continuous functions K and f vanish rapidly, for instance, K(x),f(x)=O(|x|l) near ∞ for some l<−2 and (ii) μ≥0 is sufficiently small. Especially, in the radial case with K(x)=k(|x|) and f(x)=g(|x|) for some appropriate functions k,g on [0,∞), there exist two intervals Iμ,1, Iμ,2 such that for each α∈Iμ,1 the equation has a positive entire solution uα with uα(0)=α which converges to l∈Iμ,2 at ∞, and uα1<uα2 for any α1<α2 in Iμ,1. Moreover, the map α to l is one-to-one and onto from Iμ,1 to Iμ,2. If K≥0, each solution regarded as a steady state for the corresponding parabolic equation is stable in the uniform norm; moreover, in the radial case the solutions are also weakly asymptotically stable in the weighted uniform norm with weight function |x|n−2. 相似文献
2.
Michael I Ganzburg 《Journal of Approximation Theory》1998,92(3):379-410
We determine the exact order of best approximation by polynomials and entire functions of exponential type of functions like?λ, α(x)=|x|λ exp(−A|x|−α). In particular, it is shown thatE(?λ, α, n, Lp(−1, 1))∼n−(2λp+αp+2)/2p(1+α)×exp(−(1+α−1)(Aα)1/(1+α) cos απ/2(1+α) nα/(1+α)), whereE(?λ, α, n, Lp(−1, 1)) denotes best polynomial approximation of?λ, αinLp(−1, 1),λ∈,α∈(0, 2],A>0, 1?p?∞. The problem, concerning the exact order of decrease ofE(?0, 2, n, L∞(−1, 1)), has been posed by S. N. Bernstein. 相似文献
3.
Maisa Khader 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(12):3945-3963
We study the long time behavior of solutions for damped wave equations with absorption. These equations are generally accepted as models of wave propagation in heterogeneous media with space-time dependent friction a(t,x)ut and nonlinear absorption |u|p−1u (Ikawa (2000) [17]). We consider 1<p<(n+2)/(n−2) and separable a(t,x)=λ(x)η(t) with λ(x)∼(1+|x|)−α and η(t)∼(1+t)−β satisfying conditions (A1) or (A2) which are given. The main results are precise decay estimates for the energy, L2 and Lp+1 norms of solutions. We also observe the following behavior: if α∈[0,1), β∈(−1,1) and 0<α+β<1, there are three different regions for the decay of solutions depending on p; if α∈(−∞,0) and β∈(−1,1), there are only two different regions for the decay of the solutions depending on p. 相似文献
4.
H.A. Aimar A.L. Bernardis F.J. Martín-Reyes 《Journal of Fourier Analysis and Applications》2003,9(5):497-510
We study boundedness and convergence on L
p
(R
n
,d) of the
projection operators P
j
given by MRA structures with non-necessarily
compactly supported scaling function. As a consequence, we prove that if
w is a locally integrable function such that w
-(1/p–1)(x)
(1+|x|)-N
is integrable for some N > 0, then the Muckenhoupt A
p
condition is necessary and sufficient for the associated wavelet system to
be an unconditional basis for the weighted space L
p
(R
n
,w(x) dx),
1 < p < . 相似文献
5.
Bengt-Olov Eriksson 《Journal of Approximation Theory》1998,94(3):420-454
An additive form of the Landau inequality forf∈Wn∞[−1, 1],is proved for 0<c?(cos(π/2n))−2, 1?m?n−1, with equality forf(x)=Tn(1+(x−1)/c), 1?c?(cos(π/2n))−2, whereTnis the Chebyshev polynomial. From this follows a sharp multiplicative inequality,for ‖f(n)‖?σ ‖f‖, 2n−1n! cos2n(π/2n)?σ?2n−1n!, 1?m?n−1. For these values ofσ, the result confirms Karlin's conjecture on the Landau inequality for intermediate derivatives on a finite interval. For the proof of the additive inequality a Duffin and Schaeffer-type inequality for polynomials is shown. 相似文献
6.
Xianling Fan 《Journal of Mathematical Analysis and Applications》2008,339(2):1395-1412
We study boundary trace embedding theorems for variable exponent Sobolev space W1,p(⋅)(Ω). Let Ω be an open (bounded or unbounded) domain in RN satisfying strong local Lipschitz condition. Under the hypotheses that p∈L∞(Ω), 1?infp(x)?supp(x)<N, |∇p|∈Lγ(⋅)(Ω), where γ∈L∞(Ω) and infγ(x)>N, we prove that there is a continuous boundary trace embedding W1,p(⋅)(Ω)→Lq(⋅)(∂Ω) provided q(⋅), a measurable function on ∂Ω, satisfies condition for x∈∂Ω. 相似文献
7.
Forp∈(0, 1), letQp(Qp, 0) be the space of analytic functionsfon the unit diskΔwith supw∈Δ ‖f°?w‖p<∞ (lim|w|→1 ‖f°?w‖p=0), where ‖·‖pmeans the weighted Dirichlet norm and?wis the Möbius map ofΔonto itself with?w(0)=w. In this paper, we prove the Corona theorem for the algebraQp∩H∞(Qp, 0∩H∞); then we provide a Fefferman–Stein type decomposition forQp(Qp, 0), and finally we describe the interpolating sequences forQp∩H∞(Qp, 0∩H∞)). 相似文献
8.
Lars Diening 《Mathematische Nachrichten》2004,268(1):31-43
We study the Riesz potentials Iαf on the generalized Lebesgue spaces Lp(·)(?d), where 0 < α < d and Iαf(x) ? ∫equation/tex2gif-inf-3.gif |f(y)| |x – y|α – d dy. Under the assumptions that p locally satisfies |p(x) – p(x)| ≤ C/(– ln |x – y|) and is constant outside some large ball, we prove that Iα : Lp(·)(?d) → Lp?(·)(?d), where . If p is given only on a bounded domain Ω with Lipschitz boundary we show how to extend p to on ?d such that there exists a bounded linear extension operator ? : W1,p(·)(Ω) ? (?d), while the bounds and the continuity condition of p are preserved. As an application of Riesz potentials we prove the optimal Sobolev embeddings Wk,p(·)(?d) ?Lp*(·)(Rd) with and W1,p(·)(Ω) ? Lp*(·)(Ω) for k = 1. We show compactness of the embeddings W1,p(·)(Ω) ? Lq(·)(Ω), whenever q(x) ≤ p*(x) – ε for some ε > 0. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
9.
Akihiko Inoue 《Journal of multivariate analysis》2004,89(1):135-147
Let be a fractional ARIMA(p,d,q) process with partial autocorrelation function α(·). In this paper, we prove that if d∈(−1/2,0) then |α(n)|∼|d|/n as n→∞. This extends the previous result for the case 0<d<1/2. 相似文献
10.
Zuodong Yang 《Journal of Mathematical Analysis and Applications》2003,288(2):768-783
We show the existence of entire explosive positive radial solutions for quasilinear elliptic systems div(|∇u|m−2∇u)=p(|x|)g(v), div(|∇v|n−2∇v)=q(|x|)f(u) on , where f and g are positive and non-decreasing functions on (0,∞) satisfying the Keller-Osserman condition. 相似文献
11.
A.B. Aleksandrov 《Journal of Functional Analysis》2010,258(11):3675-5251
This is a continuation of our paper [2]. We prove that for functions f in the Hölder class Λα(R) and 1<p<∞, the operator f(A)−f(B) belongs to Sp/α, whenever A and B are self-adjoint operators with A−B∈Sp. We also obtain sharp estimates for the Schatten-von Neumann norms ‖f(A)−f(B)Sp/α‖ in terms of ‖A−BSp‖ and establish similar results for other operator ideals. We also estimate Schatten-von Neumann norms of higher order differences . We prove that analogous results hold for functions on the unit circle and unitary operators and for analytic functions in the unit disk and contractions. Then we find necessary conditions on f for f(A)−f(B) to belong to Sq under the assumption that A−B∈Sp. We also obtain Schatten-von Neumann estimates for quasicommutators f(A)R−Rf(B), and introduce a spectral shift function and find a trace formula for operators of the form f(A−K)−2f(A)+f(A+K). 相似文献
12.
For any positive real numbers A, B, and d satisfying the conditions
, d>2, we construct a Gabor orthonormal basis for L2(ℝ), such that the generating function g∈L2(ℝ) satisfies the condition:∫ℝ|g(x)|2(1+|x|
A
)/log
d
(2+|x|)dx < ∞ and
. 相似文献
13.
Alan V. Lair 《Journal of Mathematical Analysis and Applications》2010,365(1):103-449
We prove that the elliptic system Δu=p(|x|)vα, Δv=q(|x|)uβ on Rn (n?3) where 0<α?1, 0<β?1, and p and q are nonnegative continuous functions has a nonnegative entire radial solution satisfying lim|x|→∞u(x)=lim|x|→∞v(x)=∞ if and only if the functions p and q satisfy
14.
For 0<p,α<∞, let ‖f‖p,α be the Lp-norm with respect the weighted measure . We define the weighted Bergman space Aαp(D) consisting of holomorphic functions f with ‖f‖p,α<∞. For any σ>0, let A−σ(D) be the space consisting of holomorphic functions f in D with . If D has C2 boundary, then we have the embedding Aαp(D)⊂A−(n+α)/p(D). We show that the condition of C2-smoothness of the boundary of D is necessary by giving a counter-example of a convex domain with C1,λ-smooth boundary for 0<λ<1 which does not satisfy the embedding. 相似文献
15.
16.
Let be a continuous function such that H(p)→H0∈R as |p|→+∞. Fixing a domain Ω in R2 we study the behaviour of a sequence (un) of approximate solutions to the H-system Δu=2H(u)ux∧uy in Ω. Assuming that supp∈R3|(H(p)−H0)p|<1, we show that the weak limit of the sequence (un) solves the H-system and un→u strongly in H1 apart from a countable set S made by isolated points. Moreover, if in addition H(p)=H0+o(1/|p|) as |p|→+∞, then in correspondence of each point of S we prove that the sequence (un) blows either an H-bubble or an H0-sphere. 相似文献
17.
Interval criteria for oscillation of second-order half-linear differential equations 总被引:1,自引:0,他引:1
By employing a generalized Riccati technique and an integral averaging method, interval oscillation criteria are established for the second-order half-linear differential equation [r(t)|x′(t)|α−1x′(t)]′+q(t)|x(t)|α−1x(t)=0. These criteria are different from most known ones in the sense that they are based on information only on a sequence of subintervals of [t0,∞), rather than on the whole half-line. They also extend, improve, and complement a number of existing results, and can be applied to extreme cases such as . In particular, several interesting examples that illustrate the importance of our results are included. 相似文献
18.
Vladimir Umanskiy 《Advances in Mathematics》2003,180(1):176-186
Given p≠0 and a positive continuous function g, with g(x+T)=g(x), for some 0<T<1 and all real x, it is shown that for suitable choice of a constant C>0 the functional has a minimizer in the class of positive functions u∈C1(R) for which u(x+T)=u(x) for all x∈R. This minimizer is used to prove the existence of a positive periodic solution y∈C2(R) of two-dimensional Lp-Minkowski problem y1−p(x)(y″(x)+y(x))=g(x), where p∉{0,2}. 相似文献
19.
Pei-Kee Lin 《Journal of Mathematical Analysis and Applications》2005,312(1):138-147
Let (X,F,μ) be a complete probability space, B a sub-σ-algebra, and Φ the probabilistic conditional expectation operator determined by B. Let K be the Banach lattice {f∈L1(X,F,μ):‖Φ(|f|)‖∞<∞} with the norm ‖f‖=‖Φ(|f|)‖∞. We prove the following theorems:
- (1)
- The closed unit ball of K contains an extreme point if and only if there is a localizing set E for B such that supp(Φ(χE))=X.
- (2)
- Suppose that there is n∈N such that f?nΦ(f) for all positive f in L∞(X,F,μ). Then K has the uniformly λ-property and every element f in the complex K with is a convex combination of at most 2n extreme points in the closed unit ball of K.
20.
Fernando Pérez-González Jouni Rättyä 《Journal of Mathematical Analysis and Applications》2009,359(2):543-57
Short proofs of the following results concerning a bounded conformal map g of the unit disc D are presented: (1) logg′ belongs to the Dirichlet space if and only if the Schwarzian derivative Sg of g satisfies Sg(z)(1−2|z|)∈L2(D); (2) logg′∈VMOA if and only if 2|Sg(z)|3(1−2|z|) is a vanishing Carleson measure on D. Analogous results for Besov and Qp,0 spaces are also given. 相似文献