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.     

Tempered fractional differential equation: variational approach

In this paper, we are concerned with the existence of ground state solution for the following fractional differential equations with tempered fractional derivative: (FD) where α∈(1/2,1), λ>0, are the left and right tempered fractional derivatives, is the fractional Sobolev spaces, and . Assuming that f satisfies the Ambrosetti–Rabinowitz condition and another suitable conditions, by using mountain pass theorem and minimization argument over Nehari manifold, we show that (FD) has a ground state solution. Furthermore, we show that this solution is a radially symmetric solution. Copyright © 2017 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.     

Asymptotics for the third‐order nonlinear Schrödinger equation in the critical case

We consider the Cauchy problem for the third‐order nonlinear Schrödinger equation where and is the Fourier transform. Our purpose in this paper is to prove the large time asymptoitic behavior of solutions for the defocusing case λ > 0 with a logarithmic correction under the non zero mass condition Copyright © 2016 John Wiley & Sons, Ltd.     

Complement of gradient estimates and Liouville theorems for nonlinear parabolic equations on noncompact Riemannian manifolds

In this paper, along the idea of Souplet and Zhang, we deduce a local elliptic‐type gradient estimates for positive solutions of the nonlinear parabolic equation: on for α ≥ 1 and α ≤ 0. As applications, related Liouville‐type theorem is exported. Our results are complement of known results. Copyright © 2016 John Wiley & Sons, Ltd.     

Global solutions to Keller‐Segel‐Navier‐Stokes equations with a class of large initial data in critical Besov spaces

In this article, we consider the Cauchy problem to Keller‐Segel equations coupled to the incompressible Navier‐Stokes equations. Using the Fourier frequency localization and the Bony paraproduct decomposition, let u_F:=e^tΔu₀; we prove that there exist 2 positive constants σ₀ and C₀ such that if the gravitational potential and the initial data (u₀,n₀,c₀) satisfy for some p,q with and , then the global solutions can be established in critical Besov spaces.     

Attractors for the strongly damped wave equation with p‐Laplacian

This paper is concerned with the initial‐boundary value problem for one‐dimensional strongly damped wave equation involving p‐Laplacian. For p > 2 , we establish the existence of weak local attractors for this problem in . Under restriction 2 < p < 4, we prove that the semigroup, generated by the considered problem, possesses a strong global attractor in , and this attractor is a bounded subset of . Copyright © 2017 John Wiley & Sons, Ltd.     

Multiple sign‐changing solutions for a class of quasilinear elliptic equations

In this paper, we study the following quasilinear Schrödinger equations: where Ω is a bounded smooth domain of , . Under some suitable conditions, we prove that this equation has three solutions of mountain pass type: one positive, one negative, and sign‐changing. Furthermore, if g is odd with respect to its second variable, this problem has infinitely many sign‐changing solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

Nonexistence results for higher order pseudo‐parabolic equations in the Heisenberg group

Sufficient conditions are obtained for the nonexistence of solutions to the nonlinear higher order pseudo‐parabolic equation where is the Kohn‐Laplace operator on the (2N + 1)‐dimensional Heisenberg group , m≥1,p > 1. Then, this result is extended to the case of a 2 × 2‐system of the same type. Our technique of proof is based on judicious choices of the test functions in the weak formulation of the sought solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

Local well‐posedness and zero‐α limit for the Euler‐α equations

In this paper, we prove the local‐in‐time existence and a blow‐up criterion of solutions in the Besov spaces for the Euler‐α equations of inviscid incompressible fluid flows in . We also establish the convergence rate of the solutions of the Euler‐α equations to the corresponding solutions of the Euler equations as the regularization parameter α approaches 0 in . Copyright © 2016 John Wiley & Sons, Ltd.     

Mehler's formulas for the univariate complex Hermite polynomials and applications

We give 2 widest Mehler's formulas for the univariate complex Hermite polynomials , by performing double summations involving the products and . They can be seen as the complex analogues of the classical Mehler's formula for the real Hermite polynomials. The proof of the first one is based on a generating function giving rise to the reproducing kernel of the generalized Bargmann space of level m. The second Mehler's formula generalizes the one appearing as a particular case of the so‐called Kibble‐Slepian formula. The proofs we present here are direct and more simpler. Moreover, direct applications are given and remarkable identities are derived.     

Wave solutions for variants of the KdV–Burger and the K(n,n)–Burger equations by the generalized G′/G‐expansion method

An application of the ‐expansion method to search for exact solutions of nonlinear partial differential equations is analyzed. This method is used for variants of the Korteweg–de Vries–Burger and the K(n,n)–Burger equations. The generalized ‐expansion method was used to construct periodic wave and solitary wave solutions of nonlinear evolution equations. This method is developed for searching exact traveling wave solutions of nonlinear partial differential equations. It is shown that the generalized ‐expansion method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear problems. Copyright © 2017 John Wiley & Sons, Ltd.     

Liouville theorems for the weighted Lane–Emden equation with finite Morse indices

In this paper, we study the nonexistence result for the weighted Lane–Emden equation: (0.1) and the weighted Lane–Emden equation with nonlinear Neumann boundary condition: (0.2) where f(|x|) and g(|x|) are the radial and continuously differential functions, is an upper half space in , and . Using the method of energy estimation and the Pohozaev identity of solution, we prove the nonexistence of the nontrivial solutions to problems 0.1 and 0.2 under appropriate assumptions on f(|x|) and g(|x|). Copyright © 2017 John Wiley & Sons, Ltd.     

The periodic solution bifurcated from homoclinic orbit for coupled ordinary differential equations

We consider the problem of the periodic solutions bifurcated from a homoclinic orbit for a pair of coupled ordinary differential equations in . Assume that the autonomous system has a degenerate homoclinic solution γ in . A functional analytic approach is used to consider the existence of periodic solution for the autonomous system with periodic perturbations. By exponential dichotomies and the method of Lyapunov–Schmidt, the bifurcation function defined between two finite dimensional subspaces is obtained, where the zeros correspond to the existence of periodic solutions for the coupled ordinary differential equations near . Copyright © 2016 John Wiley & Sons, Ltd.     

Existence of positive solutions for integral boundary value problems of fractional differential equations on infinite interval

In this paper, we consider a class of nonlinear fractional differential equations on the infinite interval with the integral boundary conditions By using Krasnoselskii fixed point theorem, the existence results of positive solutions for the boundary value problem in three cases and , are obtained, respectively. We also give out two corollaries as applications of the existence theorems and some examples to illustrate our results. Copyright © 2016 John Wiley & Sons, Ltd.     

Weak solutions to the Ginzburg–Landau model in superconductivity with the Coulomb gauge

We first prove the uniqueness of weak solutions (ψ,A) to the 3‐D Ginzburg–Landau model in superconductivity with zero magnetic diffusivity and the Coulomb gauge if , which is a critical space for some positive constant T. We also prove the global existence of solutions when and A₀∈L³. Copyright © 2016 John Wiley & Sons, Ltd.     

Two classes of asymptotically different positive solutions to advanced differential equations via two different fixed‐point principles

The paper considers a system of advanced‐type functional differential equations where F is a given functional, , r > 0 and x^t(θ) = x(t + θ), θ∈[0,r]. Two different results on the existence of solutions, with coordinates bounded above and below by the coordinates of the given vector functions if t→∞, are proved using two different fixed‐point principles. It is illustrated by examples that, applying both results simultaneously to the same equation yields two positive solutions asymptotically different for t→∞. The equation where a,τ∈(0,∞), a < 1/(τe), are constants can serve as a linear example. The existence of a pair of positive solutions asymptotically different for t→∞ is proved and their asymptotic behavior is investigated. The results are also illustrated by a nonlinear equation. Copyright © 2016 John Wiley & Sons, Ltd.     

Multiple solutions for generalized quasilinear Schrödinger equations

In this paper, we study the following generalized quasilinear Schrödinger equations: 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 positive solution, a negative solution, and a sequence of high‐energy solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

On the Hamilton–Waterloo Problem with Odd Orders

Given nonnegative integers , the Hamilton–Waterloo problem asks for a factorization of the complete graph into α ‐factors and β ‐factors. Without loss of generality, we may assume that . Clearly, v odd, , , and are necessary conditions. To date results have only been found for specific values of m and n. In this paper, we show that for any integers , these necessary conditions are sufficient when v is a multiple of and , except possibly when or 3. For the case where we show sufficiency when with some possible exceptions. We also show that when are odd integers, the lexicographic product of with the empty graph of order n has a factorization into α ‐factors and β ‐factors for every , , with some possible exceptions.     

Even Embeddings of the Complete Graphs and Their Cycle Parities

The complete graph on n vertices can be quadrangularly embedded on an orientable (resp. nonorientable) closed surface F² with Euler characteristic if and only if (resp. and ). In this article, we shall show that if quadrangulates a closed surface F², then has a quadrangular embedding on F² so that the length of each closed walk in the embedding has the parity specified by any given homomorphism , called the cycle parity.