Real‐variable characterizations of Musielak–Orlicz–Hardy spaces associated with Schrödinger operators on domains

Let n≥3, Ω be a strongly Lipschitz domain of and L_Ω:=?Δ+V a Schrödinger operator on L²(Ω) with the Dirichlet boundary condition, where Δ is the Laplace operator and the nonnegative potential V belongs to the reverse Hölder class for some q₀>n/2. Assume that the growth function satisfies that ?(x,·) is an Orlicz function, (the class of uniformly Muckenhoupt weights) and its uniformly critical lower type index , where and μ₀∈(0,1] denotes the critical regularity index of the heat kernels of the Laplace operator Δ on Ω. In this article, the authors first show that the heat kernels of L_Ω satisfy the Gaussian upper bound estimates and the Hölder continuity. The authors then introduce the ‘geometrical’ Musielak–Orlicz–Hardy space via , the Hardy space associated with on , and establish its several equivalent characterizations, respectively, in terms of the non‐tangential or the vertical maximal functions or the Lusin area functions associated with L_Ω. All the results essentially improve the known results even on Hardy spaces with p∈(n/(n + δ),1] (in this case, ?(x,t):=t^p for all x∈Ω and t∈[0,∞)). Copyright © 2016 John Wiley & Sons, Ltd.     

Existence and concentration of bound states for a Kirchhoff type problem with potentials vanishing or unbounded at infinity

In this paper, we study the existence and concentration behavior of positive solutions for the following Kirchhoff type equation: where ɛ is a positive parameter, a and b are positive constants, and 3<p<5. Let denotes the ground energy function associated with , , where is regard as a parameter. Suppose that the potential V(x) decays to zero at infinity like |x|^−α with 0<α≤2, we prove the existence of positive solutions u_ɛ belonging to for vanishing or unbounded K(x) when ɛ > 0 small. Furthermore, we show that the solution u_ɛ concentrates at the minimum points of as ɛ→0⁺.     

On the solutions of a higher‐order difference equation in terms of generalized Fibonacci sequences

This paper deals with the solutions, stability character, and asymptotic behavior of the difference equation where and the initial values x_?k,x_{?k + 1},…,x₀ are nonzero real numbers, such that their solutions are associated to Horadam numbers. Copyright © 2015 John Wiley & Sons, Ltd.     

Ground state solutions for asymptotically periodic fractional Schrödinger–Poisson problems with asymptotically cubic or super‐cubic nonlinearities

In this paper, we consider the following fractional Schrödinger–Poisson problem: where s,t∈(0,1],4s+2t>3,V(x),K(x), and f(x,u) are periodic or asymptotically periodic in x. We use the non‐Nehari manifold approach to establish the existence of the Nehari‐type ground state solutions in two cases: the periodic one and the asymptotically periodic case, by introducing weaker conditions uniformly in with and with constant θ₀∈(0,1), instead of uniformly in and the usual Nehari‐type monotonic condition on f(x,τ)/|τ|³. Our results unify both asymptotically cubic or super‐cubic nonlinearities, which are new even for s=t=1. Copyright © 2017 John Wiley & Sons, Ltd.     

Homoclinic solutions for second‐order discrete Hamiltonian systems with asymptotically quadratic potentials

In this paper, we study the existence of infinitely many homoclinic solutions for the second‐order self‐adjoint discrete Hamiltonian system , where , and are unnecessarily positive definites for all . By using the variant fountain theorem, we obtain an existence criterion to guarantee that the aforementioned system has infinitely many homoclinic solutions under the assumption that W(n,x) is asymptotically quadratic as | x | → + ∞ . Copyright © 2013 John Wiley & Sons, Ltd.     

Characterizations of Sobolev spaces associated to operators satisfying off‐diagonal estimates on balls

Let be a metric measure space of homogeneous type and L be a one‐to‐one operator of type ω on for ω ∈[0, π /2). In this article, under the assumptions that L has a bounded H _∞ ‐functional calculus on and satisfies (p _L , q _L ) off‐diagonal estimates on balls, where p _L ∈[1, 2) and q _L ∈(2, ∞ ], the authors establish a characterization of the Sobolev space , defined via L ^{α /2}, of order α ∈(0, 2] for p ∈(p _L , q _L ) by means of a quadratic function S _{α , L} . As an application, the authors show that for the degenerate elliptic operator L _w : =? w  ^? 1div(A ?) and the Schrödinger type operator with a ∈(0, ∞ ) on the weighted Euclidean space with A being real symmetric, if n ?3, with q ∈[1, 2], , p ∈(1, ∞ ) and with , then, for all , , where the implicit equivalent positive constants are independent of f , denotes the class of Muckenhoupt weights, the reverse Hölder class, and D (L _w ) and the domains of L _w and , respectively. Copyright © 2016 John Wiley & Sons, Ltd.     

Max‐type system of difference equations with positive two‐periodic sequences

This paper deals with the form and the periodicity of the solutions of the max‐type system of difference equations where , and are positive two‐periodic sequences and initial values x₀, x _{? 1}, y₀, y _{? 1} ∈ (0, + ∞ ). Copyright © 2014 John Wiley & Sons, Ltd.     

Well‐posedness and unique continuation property for the generalized Ostrovsky equation with low regularity

In this paper, we investigate the initial value problem (IVP henceforth) associated with the generalized Ostrovsky equation as follows: with initial data in the modified Sobolev space . Using Fourier restriction norm method, Tao's [k,Z]?multiplier method and the contraction mapping principle, we show that the local well‐posedness is established for the initial data with (k = 2) and is established for the initial data with (k = 3). Using these results and conservation laws, we also prove that the IVP is globally well‐posed for the initial data with s = 0(k = 2,3). Finally, using complex variables technique and Paley–Wiener theorem, we prove the unique continuation property for the IVP benefited from the ideas of Zhang ZY. et al., On the unique continuation property for the modified Kawahara equation, Advances in Mathematics (China), http://advmath.pku.edu.cn/CN/10.11845/sxjz.2014078b . Copyright © 2015 John Wiley & Sons, Ltd.     

Boundedness in a parabolic‐elliptic chemotaxis system with nonlinear diffusion and sensitivity and logistic source

In this paper, we study the zero‐flux chemotaxis‐system where Ω is a bounded and smooth domain of , n≥1, and where , k,μ>0 and α≤1. For any v≥0, the chemotactic sensitivity function is assumed to behave as the prototype χ(v)=χ₀/(1+av)², with a≥0 and χ₀>0. We prove that for any nonnegative and sufficiently regular initial data u(x,0), the corresponding initial‐boundary value problem admits a unique global bounded classical solution if α<1; indeed, for α=1, the same conclusion is obtained provided μ is large enough. Finally, we illustrate the range of dynamics present within the chemotaxis system in 1, 2, and 3 dimensions by means of numerical simulations.     

Nonlinear diffusion equations as asymptotic limits of Cahn‐Hilliard systems on unbounded domains via Cauchy's criterion

This paper develops an abstract theory for subdifferential operators to give existence and uniqueness of solutions to the initial‐boundary problem P for the nonlinear diffusion equation in an unbounded domain ( ), written as which represents the porous media, the fast diffusion equations, etc, where β is a single‐valued maximal monotone function on , and T>0. In Kurima and Yokota (J Differential Equations 2017; 263:2024‐2050 and Adv Math Sci Appl 2017; 26:221‐242) existence and uniqueness of solutions for P were directly proved under a growth condition for β even though the Stefan problem was excluded from examples of P . This paper completely removes the growth condition for β by confirming Cauchy's criterion for solutions of the following approximate problem _ε with approximate parameter ε>0: which is called the Cahn‐Hilliard system, even if ( ) is an unbounded domain. Moreover, it can be seen that the Stefan problem excluded from Kurima and Yokota (J Differential Equations 2017; 263:2024‐2050 and Adv Math Sci Appl 2017; 26:221‐242) is covered in the framework of this paper.