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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 αIμ,1 the equation has a positive entire solution uα with uα(0)=α which converges to lIμ,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.
We determine the exact order of best approximation by polynomials and entire functions of exponential type of functions like?λα(x)=|x|λ exp(−A|x|α). In particular, it is shown thatE(?λαnLp(−1, 1))∼n−(2λp+αp+2)/2p(1+α)×exp(−(1+α−1)()1/(1+α) cos απ/2(1+α) nα/(1+α)), whereE(?λαnLp(−1, 1)) denotes best polynomial approximation of?λαinLp(−1, 1),λ∈,α∈(0, 2],A>0, 1?p?∞. The problem, concerning the exact order of decrease ofE(?0, 2nL(−1, 1)), has been posed by S. N. Bernstein.  相似文献   

3.
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.
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.
An additive form of the Landau inequality forfWn[−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)‖?σf‖, 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.
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 pL(Ω), 1?infp(x)?supp(x)<N, |∇p|∈Lγ(⋅)(Ω), where γL(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 supwΔf°?wp<∞ (lim|w|→1f°?wp=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 algebraQpH(Qp, 0H); then we provide a Fefferman–Stein type decomposition forQp(Qp, 0), and finally we describe the interpolating sequences forQpH(Qp, 0H)).  相似文献   

8.
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)| |xy|αd dy. Under the assumptions that p locally satisfies |p(x) – p(x)| ≤ C/(– ln |xy|) 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.
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.
We show the existence of entire explosive positive radial solutions for quasilinear elliptic systems div(|∇u|m−2u)=p(|x|)g(v), div(|∇v|n−2v)=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.
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 ABSp. We also obtain sharp estimates for the Schatten-von Neumann norms ‖f(A)−f(B)Sp/α in terms of ‖ABSp 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 ABSp. We also obtain Schatten-von Neumann estimates for quasicommutators f(A)RRf(B), and introduce a spectral shift function and find a trace formula for operators of the form f(AK)−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:∫|g(x)|2(1+|x| A )/log d (2+|x|)dx < ∞ and .  相似文献   

13.
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
  相似文献   

14.
For 0<p,α<∞, let ‖fp,α be the Lp-norm with respect the weighted measure . We define the weighted Bergman space Aαp(D) consisting of holomorphic functions f with ‖fp,α<∞. For any σ>0, let Aσ(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)→H0R 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)uxuy in Ω. Assuming that suppR3|(H(p)−H0)p|<1, we show that the weak limit of the sequence (un) solves the H-system and unu 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.
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.
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 uC1(R) for which u(x+T)=u(x) for all xR. This minimizer is used to prove the existence of a positive periodic solution yC2(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.
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) loggVMOA 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   

20.
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 {fL1(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 nN 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.
  相似文献   

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