Periodic “weighted” operators

Consider the periodic weighted operator Ty=−ρ⁻²(ρ²y′)′ in , where the real function ρ is 1-periodic positive, and let q=ρ′/ρ∈L²(0,1). The spectrum of T consists of intervals separated by gaps γ_n,n?1, with the lengths |γ_n|. Let h_n be a height of the corresponding slit in the quasimomentum domain and let g_n,n?1, be the gap with the length |g_n| of the operator . The following results are obtained: (i) the quasimomentum k for the weighted operator T is constructed and the basic properties of k are studied, (ii) two-sided estimates of ?² norms of the sequences {|γ_n|/n}₁^∞,{h_n}₁^∞,{|g_n|}₁^∞, in terms of , (iii) the asymptotics of the gap length |γ_n| as n→∞, are determined. The proofs are based on the analysis of the quasimomentum as a conformal mapping, embedding theorems and the identities between the quasimomentum and the potential. In order to prove these results the asymptotics of the fundamental solutions and the Lyapunov function are obtained at high energy.     

Standing waves for supercritical nonlinear Schrödinger equations

Let V(x) be a non-negative, bounded potential in RN, N?3 and p supercritical, . We look for positive solutions of the standing-wave nonlinear Schrödinger equation Δu−V(x)u+up=0 in RN, with u(x)→0 as |x|→+∞. We prove that if V(x)=o(⁻²|x|) as |x|→+∞, then for N?4 and this problem admits a continuum of solutions. If in addition we have, for instance, V(x)=O(|x|^−μ) with μ>N, then this result still holds provided that N?3 and . Other conditions for solvability, involving behavior of V at ∞, are also provided.     

Positive solutions for some Schrödinger equations having partially periodic potentials

This paper is concerned with the problem of finding positive solutions of the equation −Δu+(a_∞+a(x))u=|u|^q−2u, where q is subcritical, Ω is either RN or an unbounded domain which is periodic in the first p coordinates and whose complement is contained in a cylinder , a_∞>0, a∈C(RN,R) is periodic in the first p coordinates, infx∈RN(a_∞+a(x))>0 and a(x^′,x^″)→0 as |x^″|→∞ uniformly in x^′. The cases a?0 and a?0 are considered and it is shown that, under appropriate assumptions on a, the problem has one solution in the first case and p+1 solutions in the second case when p?N−2.     

Global Strichartz estimates for the wave equation with time-periodic potentials

We obtain global Strichartz estimates for the solutions u of the wave equation for time-periodic potentials V(t,x) with compact support with respect to x. Our analysis is based on the analytic properties of the cut-off resolvent Rχ(z)=χ(U⁻¹(T)−zI)ψ₁, where U(T)=U(T,0) is the monodromy operator and T>0 the period of V(t,x). We show that if Rχ(z) has no poles z∈C, |z|?1, then for n?3, odd, we have a exponential decal of local energy. For n?2, even, we obtain also an uniform decay of local energy assuming that Rχ(z) has no poles z∈C, |z|?1, and Rχ(z) remains bounded for z in a small neighborhood of 0.     

The geometry of the critical set of nonlinear periodic Sturm-Liouville operators

We study the critical set C of the nonlinear differential operator F(u)=−u^″+f(u) defined on a Sobolev space of periodic functions Hp(S¹), p?1. Let be the plane z=0 and, for n>0, let n be the cone x²+y²=tan²z, |z−2πn|<π/2; also set . For a generic smooth nonlinearity f:R→R with surjective derivative, we show that there is a diffeomorphism between the pairs (Hp(S¹),C) and (R³,Σ)×H where H is a real separable infinite-dimensional Hilbert space.     

A generalization of the Strassen converse

This note is devoted to a generalization of the Strassen converse. Let gn:R^∞→[0,∞], n?1 be a sequence of measurable functions such that, for every n?1, and for all x,y∈R^∞, where 0<C<∞ is a constant which is independent of n. Let be a sequence of i.i.d. random variables. Assume that there exist r?1 and a function ?:[0,∞)→[0,∞) with limt→∞?(t)=∞, depending only on the sequence such that lim supn→∞gn(X₁,X₂,…)=?(Er|X|) a.s. whenever Er|X|<∞ and EX=0. We prove the converse result, namely that lim supn→∞gn(X₁,X₂,…)<∞ a.s. implies Er|X|<∞ (and EX=0 if, in addition, lim supn→∞gn(c,c,…)=∞ for all c≠0). Some applications are provided to illustrate this result.     

Heteroclinic solutions for a class of the second order Hamiltonian systems

We shall be concerned with the existence of heteroclinic orbits for the second order Hamiltonian system , where q∈Rn and V∈C¹(R×Rn,R), V?0. We will assume that V and a certain subset M⊂Rn satisfy the following conditions. M is a set of isolated points and #M?2. For every sufficiently small ε>0 there exists δ>0 such that for all (t,z)∈R×Rn, if d(z,M)?ε then −V(t,z)?δ. The integrals , z∈M, are equi-bounded and −V(t,z)→∞, as |t|→∞, uniformly on compact subsets of Rn?M. Our result states that each point in M is joined to another point in M by a solution of our system.     

Partial autocorrelation functions of the fractional ARIMA processes with negative degree of differencing

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.     

Composition operators from the Hardy space to the nth weighted-type space on the unit disk and the half-plane

The boundedness of the composition operator C_φf(z)=f(φ(z)) from the Hardy space , where X is the upper half-plane or the unit disk D={z∈C:|z|<1} in the complex plane C, to the nth weighted-type space, where φ is an analytic self-map of X, is characterized.     

Region of variability of two subclasses of univalent functions

Let F₁ (F₂ respectively) denote the class of analytic functions f in the unit disk |z|<1 with f(0)=0=f^′(0)−1 satisfying the condition RePf(z)<3/2 (RePf(z)>−1/2 respectively) in |z|<1, where Pf(z)=1+zf^″(z)/f^′(z). For any fixed z₀ in the unit disk and λ∈[0,1), we shall determine the region of variability for logf^′(z₀) when f ranges over the class and , respectively.     

Structure of boundary blow-up solutions for quasi-linear elliptic problems II: small and intermediate solutions

The structure of positive boundary blow-up solutions to quasi-linear elliptic problems of the form −Δ_pu=λf(u), u=∞ on ∂Ω, 1<p<∞, is studied in a bounded smooth domain , for a class of nonlinearities f∈C¹((0,∞)?{z₂})∩C⁰[0,∞) satisfying f(0)=f(z₁)=f(z₂)=0 with 0<z₁<z₂, f<0 in (0,z₁)∪(z₂,∞), f>0 in (z₁,z₂). Large, small and intermediate solutions are obtained for λ sufficiently large. It is known from Part I (see Structure of boundary blow-up solutions for quasilinear elliptic problems, part (I): large and small solutions, preprint), that the large solution is the unique large solution to the problem. We will see that the small solution is also the unique small solution to the problem while there are infinitely many intermediate solutions. Our results are new even for the case p=2.     

Best Approximation of Functions like |x| exp(−A|x|)

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, L_p(−1, 1))∼n^{−(2λp+αp+2)/2p(1+α)}×exp(−(1+α⁻¹)(Aα)^1/(1+α) cos απ/2(1+α) n^α/(1+α)), whereE(?_λ, α, _n, L_p(−1, 1)) denotes best polynomial approximation of?_λ, αinL_p(−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.     

Removable singularity of the polyharmonic equation

Let x₀∈Ω⊂Rⁿ, n≥2, be a domain and let m≥2. We will prove that a solution u of the polyharmonic equation Δ^mu=0 in Ω?{x₀} has a removable singularity at x₀ if and only if as |x−x₀|→0 for n≥3 and as |x−x₀|→0 for n=2. For m≥2 we will also prove that u has a removable singularity at x₀ if |u(x)|=o(|x−x₀|^2m−n) as |x−x₀|→0 for n≥3 and |u(x)|=o(|x−x₀|^2m−2log(|x−x₀|⁻¹)) as |x−x₀|→0 for n=2.     

Positive entire solutions of semilinear elliptic equations with quadratically vanishing coefficient

We establish that for n?3 and p>1, the elliptic equation Δu+K(x)up=0 in Rn possesses a continuum of positive entire solutions with logarithmic decay at ∞, provided that a locally Hölder continuous function K?0 in Rn?{0}, satisfies K(x)=O(σ|x|) at x=0 for some σ>−2, and ²|x|K(x)=c+O([log|x|]^−θ) near ∞ for some constants c>0 and θ>1. The continuum contains at least countably many solutions among which any two do not intersect. This is an affirmative answer to an open question raised in [S. Bae, T.K. Chang, On a class of semilinear elliptic equations in Rn, J. Differential Equations 185 (2002) 225-250]. The crucial observation is that in the radial case of K(r)=K(|x|), two fundamental weights, and , appear in analyzing the asymptotic behavior of solutions.     

Note on a Fermat-type diophantine equation

Let p>5 be a prime number and ζ a pth root of unity. Let c be an integer divisible only by primes of the form kp−1,(k,p)=1.Let C_p⁽ⁱ⁾ be the eigenspace of the p-Sylow subgroup of ideal class group C of corresponding to ωi,ω being the Teichmuller character.In this article we extend the main theorem in Sitaraman (J. Number Theory 80 (2000) 174) and get the following: For any fixed odd positive integer n<p−4, assume:

(a)

At least one of C_p⁽³⁾,C_p⁽⁵⁾,…,C_p⁽ⁿ⁾ is non-trivial.

(b)

C_p⁽ⁱ⁾=0 for p−n−1?i?p−2.

(c)

for 1?i?n+1.

Let q be an odd prime such that , and such that there is a prime ideal Q over q in whose ideal class is of the form I^pJ where J is non-trivial, not a pth power and J∈C_p⁽³⁾⊕C_p⁽⁵⁾⊕?⊕C_p⁽ⁿ⁾.For such p and q, if x^p+y^p=pcz^p has a non-trivial solution , with (x,y,z)=1, then .Let t(n)=n²²⁴ⁿ⁴. If , then applying a result of Soulé (J. Reine Angew. Math. 517 (1999) 209), we show that the above result holds with only condition (a) because the others are automatically satisfied.We also make a remark about the effect of Soulé's result on the p-divisibility of h_p⁺ (the class number of the maximal real subgroup of ) which is relevant to the existence of integral solutions to x^p+y^p=pcz^p.