Pseudo–almost‐periodic solutions for delayed differential equations with integrable dichotomies and bi–almost‐periodic Green functions

Using the existence of integrable bi–almost‐periodic Green functions of linear homogeneous differential equations and the contraction fixed point, we are able to prove the existence of almost and pseudo–almost‐periodic mild solutions under quite general hypotheses for the differential equation with constant delay in a Banach space X, where τ>0 is a fixed constant. The results extend the corresponding ones in the case of exponential dichotomy. Some examples illustrate the importance of the concepts.     

On the existence of positive least energy solutions for a coupled Schrödinger system with critical exponent

In this paper, we consider the following coupled Schrödinger system with critical exponent: where is a smooth bounded domain, λ > 0,μ≥0, and . Under certain conditions on λ and μ, we show that this problem has at least one positive least energy solution. Copyright © 2016 John Wiley & Sons, Ltd.     

Existence and multiplicity of systems of Kirchhoff‐type equations with general potentials

This paper is concerned with the following systems of Kirchhoff‐type equations: Under more relaxed assumptions on V(x) and F(x,u,v), we first prove the existence of at least two nontrivial solutions for the aforementioned system by using Morse theory in combination with local linking arguments. Then by using the Clark theorem, the existence results of at least 2k distinct pairs of solutions are obtained. Some recent results from the literature are extended. Copyright © 2016 John Wiley & Sons, Ltd.     

Ground state solutions and geometrically distinct solutions for generalized quasilinear Schrödinger equation

In this paper, we study the following generalized quasilinear Schrödinger equation where N≥3, is a C¹ even function, g(0) = 1 and g^′(s) > 0 for all s > 0. Under some suitable conditions, we prove that the equation has a ground state solution and infinitely many pairs ±u of geometrically distinct solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

Existence and concentration of positive solutions for coupled Schrödinger equations

In this paper, we study the existence and concentration of positive solution of a class of coupled Schrödinger equations. We admit that the potentials may not be non-negative and suppose that the intersection of the sets has positive Lebesgue measure. By studying the modified functional of the associated functional carefully, we establish the existence of positive least energy solutions for the coupled Schrödinger system. Moreover, we prove the concentration phenomenon of the positive solution when the parameter goes to infinity.     

The global weak solution for the shallow water wave model of moderate amplitude

In this paper, we firstly establish the existence theorem of the global weak solutions of the Cauchy problem for the shallow water wave model of moderate amplitude, then following the idea in Xin and Zhang’s work (see Xin and Zhang, 2002), we prove the uniqueness of global weak solutions using the localization analysis in the transport equation theory. Finally, several travelling wave solutions are derived.     

Micro‐ and Nano‐Liquid Phases Coexistent with Ice as Separation and Reaction Media.

Ice has a variety of scientifically interesting features, some of which have not been reasonably interpreted despite substantial efforts by researchers. Most chemical studies of ice have focused on the elucidation of its physicochemical nature and its roles in the natural environment. Ice often contains impurities, such as salts, and in such cases, a liquid phase coexists with solid ice over a wide temperature range. This impure ice also acts as a cryoreactor, governing the circulation of chemical species of environmental importance. Reactions and phenomena occurring in this liquid phase show features different from those seen in normal bulk aqueous solutions. In the present account, we discuss the chemical characteristics of the liquid phase that develops in a frozen aqueous phase and show how novel analytical systems can be designed based on he features of the liquid phase which are predictable in some cases but unpredictable in others.     

Nonlocal transmission problems with fractional diffusion and boundary conditions on non-smooth interfaces

We consider a transmission problem consisting of two semilinear parabolic equations involving fractional diffusion operators of different orders in a general non-smooth setting with emphasis on Lipschitz interfaces and transmission conditions along the interface. We give a unified framework for the existence and uniqueness of strong and mild solutions, and their global regularity properties.