Multiple solutions for a modified Kirchhoff‐type equation in RN

In this paper, we study the following modified Kirchhoff‐type equations of the form: where a > 0, b ≥ 0, and . Under appropriate assumptions on V (x) and h(x,u), some existence results for positive solutions, negative solutions, and sequence of high energy solutions are obtained via a perturbation method. Copyright © 2014 John Wiley & Sons, Ltd.     

Infinitely many solutions for a Hardy–Sobolev equation involving critical growth

In this paper, by an approximating argument, we obtain infinitely many solutions for the following Hardy–Sobolev equation with critical growth: provided N > 6 + t, where and Ω is an open bounded domain in , which contains some points x₀ = (0,z₀). Copyright © 2013 John Wiley & Sons, Ltd.     

Boundedness of solutions to parabolic–elliptic Keller–Segel systems with signal‐dependent sensitivity

This paper deals with the parabolic–elliptic Keller–Segel system with signal‐dependent chemotactic sensitivity function, under homogeneous Neumann boundary conditions in a smooth bounded domain , with initial data satisfying u₀ ≥ 0 and . The chemotactic sensitivity function χ(v) is assumed to satisfy The global existence of weak solutions in the special case is shown by Biler (Adv. Math. Sci. Appl. 1999; 9:347–359). Uniform boundedness and blow‐up of radial solutions are studied by Nagai and Senba (Adv. Math. Sci. Appl. 1998; 8:145–156). However, the global existence and uniform boundedness of classical nonradial solutions are left as an open problem. This paper gives an answer to the problem. Namely, it is shown that the system possesses a unique global classical solution that is uniformly bounded if , where γ > 0 is a constant depending on Ω and u₀. Copyright © 2014 John Wiley & Sons, Ltd.     

Infinitely many solutions to elliptic systems involving critical exponents and Hardy potential

In this paper, we consider the following elliptic systems involving critical Sobolev growth and Hardy potential: where N ≥ 3,λ₁,λ₂ ∈ [0,Λ_N), is the best Hardy constant. is the critical Sobolev exponent. a₁,a₂, b are positive parameters, α,β > 0 and 1 < α + β : = q < 2 < 2*. with . By means of the concentration‐compactness principle and Kajikiya's new version of symmetric mountain pass lemma, we obtain infinitely many solutions that tend to zero for suitable positive parameters a₁,a₂,b and λ₁,λ₂. Copyright © 2012 John Wiley & Sons, Ltd.     

On the Hartley – Fourier sine generalized convolution

In this paper, we construct and study a new generalized convolution (f * g)(x) of functions f,g for the Hartley (H₁,H₂) and the Fourier sine (F_s) integral transforms. We will show that these generalized convolutions satisfy the following factorization equalities: We prove the existence of this generalized convolution on different function spaces, such as . As examples, applications to solve a type of integral equations and a type of systems of integral equations are presented. Copyright © 2013 John Wiley & Sons, Ltd.     

Non‐autonomous homogeneous rational difference equations of degree one: convergence and monotone solutions for second and third order case

Consider the non‐autonomous equations: where and also These are some non‐autonomous homogeneous rational difference equations of degree one. A reduction in order is consi‐dered. Convergence and monoton character of positive solutions are studied. Copyright © 2013 John Wiley & Sons, Ltd.     

A new Beale–Kato–Majda criteria for the 3D magneto‐micropolar fluid equations in the Orlicz–Morrey space

In this work, we are interested in the study of regularity for the three‐dimensional magneto‐micropolar fluid equations in Orlicz–Morrey spaces. If the velocity field satisfies or the gradient field of velocity satisfies then we show that the solution remains smooth on [0,T]. In view of the embedding with 2 < p < 3 ∕ r and P > 1, we see that our result extends the result of Yuan and that of Gala. Copyright © 2012 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.     

Hölder regularity of the gradient for the non‐homogeneous parabolic p(x,t)‐Laplacian equations

In this paper, we obtain the local Hölder regularity of the gradient of weak solutions for the non‐homogeneous parabolic p(x,t)‐Laplacian equations provided p(x,t), A and f are Hölder continuous functions. Copyright © 2013 John Wiley & Sons, Ltd.     

The Bergman–Sce transform for slice monogenic functions

In a recent paper, we showed that the classical Bergman theory admits two possible formulations for the class of slice regular functions with quaternionic values. In the so called formulation of the first kind, we provide a Bergman kernel which is defined on and is a reproducing kernel. In the so called formulation of the second kind, we use the Representation Formula for slice regular functions to define a second Bergman kernel; this time the kernel is still defined on U, but the integral representation of f is based on an integral computed only on and the integral does not depend on , (here denotes the sphere unit of purely imaginary quaternions, and represents the complex plane with imaginary unit I). In this paper, we extend the second formulation of the Bergman theory to the case of slice monogenic functions and we focus our attention on the so‐called Bergman–Sce transform. This integral transform is defined by using the Bergman kernel and the Sce mapping theorem and associates to every slice monogenic function f, an axially monogenic function . Copyright © 2011 John Wiley & Sons, Ltd.     

A remark on the logarithmically improved regularity criterion for the micropolar fluid equations in terms of the pressure

In this paper, we study the regularity criterion of weak solutions of the three dimensional micropolar fluid flows. It is proved that if the pressure satisfies where denotes the critical Besov space, then the weak solution (u,w) becomes a regular solution on (0,T]. This regularity criterion can be regarded as log in time improvements of the standard Serrin's criteria established before. Copyright © 2011 John Wiley & Sons, Ltd.     

Existence of positive solutions to a coupled system of fractional differential equations

We consider the following system of fractional differential equations where is the Riemann‐Liouville fractional derivative of order α,f,g : [0,1] × [0, ∞ ) × [0, ∞ ) → [0, ∞ ). Sufficient conditions are provided for the existence of positive solutions to the considered problem. Copyright © 2014 John Wiley & Sons, Ltd.     

A semilinear heat equation with a localized nonlinear source and non‐continuous initial data

This paper is devoted to the study of the Cauchy problem for a semilinear heat equation with nonlinear term presenting a nonlinear source centered in a closed region of the spatial domain Ω. We assume that is either a smooth bounded domain or the whole space , The initial data is assumed to belong to the Lebesgue space . Copyright © 2011 John Wiley & Sons, Ltd.     

The asymptotic behavior of Caputo delta fractional equations

Consider the following ν‐th order Caputo delta fractional equation (0.1) The following asymptotic results are obtained. Theorem A. Assume 0 < ν < 1 and there exists a constant b₂ such that c(t)≥b₂>0. Then the solutions of the equation (0.1) with x(a) > 0 satisfy      

Solutions for perturbed biharmonic equations with critical nonlinearity

In this paper, we study the perturbed biharmonic equations where Δ² is the biharmonic operator, is the Sobolev critical exponent, p ∈ (2,2 ^{* *} ), P(x), and Q(x) are bounded positive functions. Under some given conditions on V, we prove that the problem has at least one nontrivial solution provided that and that for any , it has at least n ^* pairs solutions if , where and are sufficiently small positive numbers. Moreover, these solutions u_ε → 0 in as ε → 0. Copyright © 2013 The authors. Mathematical Methods in the Applied Sciences published by John Wiley & Sons, Ltd.     

Infinitely many peak solutions for a biharmonic equation involving critical exponent

In this paper, we study the following biharmonic equation where , K(1) > 0,K′(1) > 0, B₁(0) is the unit ball in (N≥6). We show that the aforementioned problem has infinitely many peak solutions, whose energy can be made arbitrarily large. 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.     

Simple eigenvalues for the Steklov problem in a domain with a small hole. A functional analytic approach

Let be a bounded open domain of . Let denote the outward unit normal of . We assume that the Steklov problem Δu = 0 in and on has a simple eigenvalue of . Then we consider an annular domain obtained by removing from a small‐cavity size of ε > 0, and we show that under proper assumptions there exists a real valued and real analytic function defined in an open neighborhood of (0,0) in and such that is a simple eigenvalue for the Steklov problem Δu = 0 in and on for all ε > 0 small enough, and such that . Here denotes the outward unit normal of , and δ_2,2 ≡ 1 and δ_2,n ≡ 0 if n ≥ 3. Then related statements have been proved for corresponding eigenfunctions. Copyright © 2013 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.     

Spectral asymptotics and basis properties of fourth order differential operators with regular boundary conditions

In this paper, we consider the problem where λ is a spectral parameter; q(x) ∈ L₁(0,1) is complex‐valued function; α_s, s = 1,2,3, are arbitrary complex constants that satisfy α₂ = α₁ + α₃ and σ = 0,1. The boundary conditions of this problem are regular, but not strongly regular. Asymptotic formulae for eigenvalues and eigenfunctions of the considered boundary value problem are established. It is proved that all the eigenvalues, except for finite number, are simple and the system of root functions of this spectral problem forms a basis in the space L_p(0,1), 1 < p < ∞ , when ; moreover, this basis is unconditional for p = 2. We note that the considered problem was previously investigated in the condition of α₂ ≠ α₁ + α₃. Copyright © 2013 John Wiley & Sons, Ltd.