EXISTENCE AND BLOW-UP BEHAVIOR OF CONSTRAINED MINIMIZERS FOR SCHRDINGER-POISSON-SLATER SYSTEM

In this article,we study constrained minimizers of the following variational problem e(p):=inf{u∈H1(R3),||u||22=p}E(u),p〉0,where E(u)is the Schrdinger-Poisson-Slater(SPS)energy functional E(u):=1/2∫R3︱▽u(x)︱2dx-1/4∫R3∫R3u2(y)u2(x)/︱x-y︱dydx-1/p∫R3︱u(x)︱pdx in R3 and p∈(2,6).We prove the existence of minimizers for the cases 2p10/3,ρ0,and p=10/3,0ρρ~*,and show that e(ρ)=-∞for the other cases,whereρ~*=||φ||_2~2 andφ(x)is the unique(up to translations)positive radially symmetric solution of-△u+u=u~(7/3)in R~3.Moreover,when e(ρ~*)=-∞,the blow-up behavior of minimizers asρ↗ρ~*is also analyzed rigorously.     

MULTIPLICITY RESULTS FOR FOURTH ORDER ELLIPTIC EQUATIONS OF KIRCHHOFF-TYPE

In this paper, we concern with the following fourth order elliptic equations of Kirchhoff type {△2u- a+b∫R3|▽u|2dx △u+V(x)u=f(x, u), x∈ R3,u∈H2(R3),wherea, b 0 are constants and the primitive of the nonlinearityfis of superlinear growth near infinity inuand is also allowed to be sign-changing. By using variational methods, we establish the existence and multiplicity of solutions. Our conditions weaken the AmbrosettiRabinowitz type condition.     

The local and global existence of solutions for a generalized Camassa-Holm equation

A fully nonlinear generalization of the Camassa-Holm equation is investigated. Using the pseudoparabolic regularization technique, its local well-posedness in Sobolev space Hs(R) with s3/2 is established via a limiting procedure. Provided that the initial momentum (1-x2)u0 satisfies the sign condition, u0∈Hs(R)(s3/2) and u0∈L1(R),the existence and uniqueness of global solutions for the equation are shown to be true in the space C([0,∞); Hs(R))∩C1([0,∞);Hs-1(R)).     

RESEARCHANNOUNCEMENTS——TheCauchyProblemsandGlobalSolutionstoDeybeSystemandBurgers''''EquationsinPseudomeasureSpaces

In this paper we consider the construction of solutions to the Cauchy problem of Burgers' equationsut-γ△u + u·▽u = 0, t∈R+,x∈R3, (1)u(0,x)= u0(x), x ∈ R3, (2)in pseudomeasure spaces, where γ(?) 0 is a small parameter that plays the role of the viscosity and u = u(t,x) is a velocity-like vector field defined on R+×R3. The initial datum u0(x) is a vector-valued function defined on R3, In one space dimension Burgers' equation is     

Remarks on the Regularity Criteria of Three-Dimensional Navier-Stokes Equations in Margin Case

In the study of the regularity criteria for Leray weak solutions to threedimensional Navier-Stokes equations, two sufficient conditions such that the horizontal velocity u satisfies u∈L2（0,T;BMO（R3）） or u∈L^2/1＋r（0,T;B∞,∞（R3）） for 0 〈 r 〈 1 are considered.     

A remark on the generalized second order Toda system

In this paper, we establish a priori estimates to the generalized second order Toda system{-Δu1(x)=2R1(x)eu1-R2eu2,-Δu2(x)=R1(x)eu1+2R2eu2 in R2 , and discuss the convergence and asymptotic behavior of its solutions, where Ri(x), i=1, 2, is bounded function in R2 . Consequently, we prove that all the solutions satisfy an identity, which is somewhat a generalization of the well-known Kazdan-Warner condition.     

A Note on Commutators with Power Central Values on Lie Ideals

This note proves that, if R is a prime ring of characteristic 2 with d a derivation of R and L a noncentral Lie ideal of R such that [d（u）,u]^n is central, for all u ∈ L, then R must satisfy s4, the standard identity in 4 variables. The case where R is a semiprime ring is also examined by the authors. The results of the note improve Carini and Filippis＇s results.     

NON-UNIFORM DEPENDENCE ON INITIAL DATA FOR THE MODIFIED CAMASSA-HOLM EQUATION ON THE LINE

In this paper, we study the Cauchy problem for the modified Camassa-Holm equation mt + umx + 2ux m = 0, m =(1- δ2x)2u,u(x, 0) = u0(x) ∈ Hs(R), x ∈ R, t 0,and show that the solution map is not uniformly continuous in Sobolev spaces Hs(R) for s 7/2. Compared with the periodic problem, the non-periodic problem is more difficult,e.g., it depends on the conservation law. Our proof is based on the estimates for the actual solutions and the approximate solutions, which consist of a low frequency and a high frequency part.     

Infinitely many radial solutions to elliptic problems with critical Sobolev and Hardy terms

Let Ω RN be a ball centered at the origin with radius R > 0 and N 7, 2* = 2N/N-2. We obtain the existence of infinitely many radial solutions for the Dirichlet problem -△u = μ |x|2 u |u|2*-2u λu in Ω, u = 0 on аΩ for suitable positive numbers μ and λ. Such solutions are characterized by the number of their nodes.     

Homo－＆ Heteroclinic Orbits in Conservative Systems With 3－wells Potential

Consider the following 2×2 nonlinear system:where f(u): R→R is a, smooth function. Setwhere F'(u)= f(u). Then (1) can be rewritten as an equivalent Hamiltonian system:     

On periodic solutions of even-order ordinary differential equations

Optimal in a certain sense sufficient conditions are given for the existence and uniqueness of ω-periodic solutions of the nonautonomous ordinary differential equation u ^(2m) =f(t,u,...,u ^(m-1) ), where the function f:ℝ×ℝ ^m →ℝ is periodic with respect to the first argument with period ω. Received: December 21, 1999; in final form: August 12, 2000?Published online: October 2, 2001     

An elliptic continuation result for harmonic functions in two dimensions

Let u be harmonic in a simply connected domainG ⊂ ℝ² and letK be a compact subset of G. In this note, it is proved there exists an “elliptic continuation” of u, namely there exist a smooth functionu ₁ and a second order uniformly elliptic operatorL with smooth coefficients in ℝ², satisfying:u ₁=u inK, Lu ₁=0 in ℝ². A similar continuation theorem, with u itself a solution to an elliptic second order equation inG, is also proved.     

Existence principles for higher-order nonlocal boundary-value problems and their applications to singular Sturm-Liouville problems

We present existence principles for the nonlocal boundary-value problem (φ(u^(p−1)))′=g(t,u,...,u^(p−1), α_k(u)=0, 1≤k≤p−1, where p ≥ 2, π: ℝ → ℝ is an increasing and odd homeomorphism, g is a Carathéodory function that is either regular or has singularities in its space variables, and α _k: C ^p−1[0, T] → ℝ is a continuous functional. An application of the existence principles to singular Sturm-Liouville problems (−1)ⁿ(φ(u⁽²ⁿ⁻⁾))′=f(t,u,...,u⁽²ⁿ⁻¹⁾), u^(2k)(0)=0, α_ku^(2k)(T)+b_ku^(2k=1)(T)=0, 0≤k≤n−1, is given. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 2, pp. 240–259, February, 2008.     

Asymptotic behavior of solutions to a wave equation with a non linear dissipative term in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this note we investigate the asymptotic behavior of solutions to the wave equation:u"-Δu+g(u')=0 in ℝⁿxℝ₊.     

Differential Invariants of the 2D Conformal Lie Algebra Action

We consider the Lie algebra that corresponds to the Lie pseudogroup of all conformal transformations on the plane. This conformal Lie algebra is canonically represented as the Lie algebra of holomorphic vector fields in ℝ²≃ℂ. We describe all representations of \mathfrakg\mathfrak{g} via vector fields in J ⁰ℝ²=ℝ³(x,y,u), which project to the canonical representation, and find their algebra of scalar differential invariants.     

Kernel regression estimation for continuous spatial processes

We investigate here a kernel estimate of the spatial regression function r(x) = E(Y _u | X _u = x), x ∈ ℝ^d, of a stationary multidimensional spatial process { Z _u = (X _u, Y _u), u ∈ ℝ ^N }. The weak and strong consistency of the estimate is shown under sufficient conditions on the mixing coefficients and the bandwidth, when the process is observed over a rectangular domain of ℝ^N. Special attention is paid to achieve optimal and suroptimal strong rates of convergence. It is also shown that this suroptimal rate is preserved by using a suitable spatial sampling scheme.      

Existence of T-solution for degenerated problem via Minty’s lemma

We study the existence result of solutions for the nonlinear degenerated elliptic problem of the form, -div（a（x, u,△↓u）） = F in Ω, where Ω is a bounded domain of R^N, N≥2, a ：Ω×R×R^N→R^N is a Carathéodory function satisfying the natural growth condition and the coercivity condition, but they verify only the large monotonicity. The second term F belongs to W^-1,p′（Ω, w^＊）. The existence result is proved by using the L^1-version of Minty＇s lemma.     

An Almost Sure Limit Theorem for Super-Brownian Motion

We establish an almost sure scaling limit theorem for super-Brownian motion on ℝ ^d associated with the semi-linear equation u_t=\frac12Du+bu-au²u_{t}=\frac{1}{2}\Delta u+\beta u-\alpha u^{2} , where α and β are positive constants. In this case, the spectral theoretical assumptions required in Chen et al. (J. Funct. Anal. 254:1988–2019, 2008) are not satisfied. An example is given to show that the main results also hold for some sub-domains in ℝ ^d .     

Partial regularity for weak solutions of a nonlinear elliptic equation

For scalar non-linear elliptic equations, stationary solutions are defined to be critical points of a functional with respect to the variations of the domain. We consideru a weak positive solution of −Δu=u ^α in -Δu=u ^α in Ω ⊂ ℝ ⁿ , which is stationary. We prove that the Hausdorff dimension of the singular set ofu is less thann−2α+1/α−1, if α≥n+2/n−2.     

Removable singularities and non-uniqueness of solutions of a singular diffusion equation

 We prove that the solution u of the equation u _t =Δlog u, u>0, in (Ω\{x ₀})×(0,T), Ω⊂ℝ², has removable singularities at {x ₀}×(0,T) if and only if for any 0<α<1, 0<a<b<T, there exist constants ρ₀, C ₁, C ₂>0, such that C ₁ |x−x ₀|^α≤u(x,t)≤C ₂|x−x ₀|^−α holds for all 0<|x−x ₀|≤ρ₀ and a≤t≤b. As a consequence we obtain a sufficient condition for removable singularities at {∞}×(0,T) for solutions of the above equation in ℝ²×(0,T) and we prove the existence of infinitely many finite mass solutions for the equation in ℝ²×(0,T) when 0≤u ₀∉L ¹ (ℝ²) is radially symmetric and u ₀L _loc ¹(ℝ²). Received: 16 December 2001 / Revised version: 20 May 2002 / Published online: 10 February 2003 Mathematics Subject Classification (1991): 35B40, 35B25, 35K55, 35K65