bluuebunny/crossref_metadata_2025 huggingface dataset:
Generalized Attractor-Repeller Pairs, Diagonalizability and Integral Separation
Authors: Palmer Ken , Siegmund Stefan
Published: March 2016

We introduce the notion of a generalized attractor-repeller pair for ẋ = A(t)x which in fact is an attractor-repeller pair for the induced (nonlinear) flow on the projective space, and use it to characterize diagonalizability. We show that integral separation implies the existence of generalized attractor-repeller pairs, and use topological dynamics, more precisely the hull of A(t), to prove a converse theorem. The relation to exponential separation is also discussed.
On the Number of Solutions to Semilinear Boundary Value Problems
Authors: Kunze Markus , Ortega Rafael
Published: March 2016

We consider semilinear elliptic problems of the form Δu + g(u) = f(x) with Neumann boundary conditions or Δu+λ1u+g(u) = f(x) with Dirichlet boundary conditions, and we derive conditions on g and f under which an upper bound on the number of solutions can be obtained.
On Generalized and Viscosity Solutions of Nonlinear Elliptic Equations
Authors: Hartenstine David , Schmitt Klaus
Published: March 2016

There are many notions of solutions of nonlinear elliptic partial differential equations. This paper is concerned with solutions which are obtained as suprema (or infima) of so-called subfunctions (superfunctions) or viscosity subsolutions (viscosity supersolutions). The paper also explores the relationship of these (generalized) solutions of differential inequalities and provides a relevant example for which existence questions have been studied using these concepts.
On Some Results of Moser and of Bangert, II
Authors: Rabinowitz Paul H. , Stredulinsky Ed
Published: March 2016

Existence is proved for heteroclinic solutions of a family of semilinear elliptic PDE’s on ℝn treated earlier by Bangert [2]. A variational framework for finding solutions of this problem is presented. It will be used in a sequel to construct more complicated homoclinic and heteroclinic solutions of the equation.
Asymptotic Profile for a Fourth Order PDE with Critical Exponential Growth in Dimension Four
Authors: Robert Frédéric , Struwe Michael
Published: March 2016

Let Ω be a smooth domain of R4. In this paper, we consider a family (u∊)∊>0 of positive solutions in C4(Ω̅) to the equation

where λ ∈ ℝ. Assuming that u∊→ 0 weakly in H2
2,0(Ω) while supΩ u∊ →∞, we describe the asymptotics of u∊ as ∊ → 0.
Homology Classes of the Circle Space on Spheres and the Discontinuity of Deformations
Authors: Zhou Congyi , Longy Yiming
Published: March 2016

The famous Lusternik-Schnirelmann theorem claims that on a surface of genus 0 there exist at least three simple closed geodesics without self-intersections. A variational proof of this theorem is given in the book “Riemannian Geometry (2ed Ed.)” of W. Klingenberg. In this paper, firstly we point out that the construction of the three non-trivial relative homology classes in the circle space on S2 in this proof (the proof of Proposition 3.7.19 of [7]) is incorrect, and give explicit con- structions of these homology classes. Secondly, we construct a counter-example to show that the deformation constructed in this proof in the closure of non- self-intersecting geodesic polygons (on pp.343-344 of [7]) is discontinuous, and therefore this proof is not complete.
Necessary and Sufficient Conditions for Extinction of One Species
Authors: Tineo Antonio
Published: March 2016

In this paper we consider n+1 competing species such that n of them are permanent and we study the behavior of the rest of the species. Our study complements a work initiated by S. Ahmad and A.C. Lazer concerning the extinction or persistence of the rest species.
Qualitative Properties of Solutions of m-Laplace Systems
Authors: Damascelli Lucio , Sciunzi Berardino
Published: March 2016

We prove regularity results for the solutions of the equation -Δmu = h(x), such as summability properties of the second derivatives and summability properties of

. Analogous results were recently proved by the authors for the equation -Δmu = f(u). These results allow us to extend to the case of systems of m-Laplace equations, some results recently proved by the authors for the case of a single equation. More precisely we consider the problem

and we prove regularity properties of the solutions as well as qualitative properties of the solutions. Moreover we get a geometric characterization of the critical sets Zu ≡ {x ∈ Ω | Du(x) = 0} and Zv ≡ Dv(x) = 0. In particular we prove that in convex and symmetric domains we have Zu = {0} = Zv, assuming that 0 is the center of symmetry.
A Denoised Version of Some Kinds of Set Convergence
Authors: Lops Filomena A.
Published: March 2016

The aim of this paper consists of introducing on a locally compact and σ-compact metric space a notion of set convergence, which generalizes the Hausdorff convergence, the local Hausdorff convergence and the Kuratowski convergence. We analyze the connections beetwen the three new notions: and. in particular, we prove a compactness result. As a first application of this convergence we give, on a sequence of sets, a condition which assures the lower semicontinuity of the Hausdorff measure with respect to this new convergence and we show that this condition is satisfied by any minimizing sequence of Mumford-Shah functional.
Averaging of Attractors and Inertial Manifolds for Parabolic PDE With Random Coefficients
Authors: Chueshov Igor D. , Schmalfuß Björn
Published: March 2016

The averaging method has been used to study random or non-autonomous systems on a fast time scale. We apply this method to a random abstract evolution equation on a fast time scale whose long time behavior can be characterized by a random attractor or a random inertial manifold. The main purpose is to show that the long-time behavior of such a system can be described by a deterministic evolution equation with averaged coefficients. Our first result provides an averaging result on finite time intervals which we use to show that under a dissipativity assumption the attractors of the fast time scale systems are upper semicontinuous when the scaling parameter goes to zero. Our main result deals with a global averaging procedure. Under some spectral gap condition we show that inertial manifolds of the fast time scale system tend to an inertial manifold of the averaged system when the scaling parameter goes to zero. These general results can be applied to semilinear parabolic differential equations containing a scaled ergodic noise on a fast time scale which includes scaled almost periodic motions.
Persistence for a Class of Triangular Cross Diffusion Parabolic Systems
Authors: Le Dung , Nguyen Toan Trong
Published: March 2016

The purpose of this paper is to investigate the dynamics of a class of triangular parabolic systems given on bounded domains of arbitrary dimension. In particular, the existence of global attractors and the persistence property will be established.
The Radial Solutions of Monge-Ampère Equations and the Semi-geostrophic System
Authors: Chen Xiuqing , Jian Huaiyu
Published: March 2016

We first obtain an (optimal) regularity condition for entire radial solutions of Monge- Ampère equations and then study the existence of radial solutions of the Semi-geostrophic system which is coupled by a Monge-Ampère and a transport equation.
Some Elliptic Problems With Degenerate Coercivity
Authors: Boccardo Lucio
Published: March 2016

In this paper we are interested in existence of solutions for some nonlinear elliptic equations with principal part having degenerate coercivity. The model case is

with Ω bounded open subset of ℝN, N ≥ 2 and f a datum in some Lebesgue space.
Periodic Solutions of Symmetric Elliptic Singular Systems: the Higher Codimension Case
Authors: Battelli Flaviano , Fečkan Michal
Published: March 2016

We show the existence of periodic solutions of certain singularly perturbed systems having symmetry properties. Our result applies to some singular systems arising in the study of Hamiltonian systems with a strong restoring force and extends to a higher codimension the result obtained in our previous paper [4].
A Note on Nonlinear Schrödinger Systems: Existence of A-symmetric Solutions
Authors: Ambrosetti Antonio
Published: March 2016

The paper contains some results concerning the existence of a-symmetric solutions to a class of autonomous nonlinear Schrödinger systems, coupled through a small parameter.
Multiple Solitary Waves For a Non-homogeneous Schrödinger–Maxwell System in ℝ3
Authors: Salvatore A.
Published: March 2016

We look for standing waves of nonlinear Schrödinger equation

coupled with Maxwell’s equations. We use the variational formulation introduced by Benci and Fortunato in 1992 for studying an eigenvalue problem for the Schrödinger-Maxwell system in bounded domains. We establish the existence of multiple standing waves both in the homogeneous and the non-homogeneous cases by means of the fibering method introduced by Pohozaev.
Semilinear Elliptic Systems With Exponential Nonlinearities in Two Dimensions
Authors: Figueiredo Djairo G. de , Ó João Marcos do , Ruf Bernhard
Published: March 2016

We establish a priori bounds for positive solutions of semilinear elliptic systems of the form

where Ω is a bounded and smooth domain in ℝ2. We obtain results concerning such bounds when f and g depend exponentially with respect to u and v.
Multiple Solutions of Some Nonlinear Variational Problems
Authors: Candela A.M. , Palmieri G.
Published: March 2016

The aim of this paper is to prove some existence and multiplicity results for functionals of type J(u) = ∫Ω­ A(x, u)|▽u|2dx - ∫Ω­ G(x, u)dx, u 2 D ∊ H0
1 (Ω), with bounded domain Ω in ℝN. Since, in general, J is not Gâteaux differentiable in D, we study its restriction on the Banach space X = H0
1 (Ω) ∩ L∞(Ω) and apply some abstract existence and multiplicity theorems involving a variant of condition (C) below.
Normalized Solutions to Strongly Indefinite Semilinear Equations
Authors: Buffoni Boris , Esteban Maria J. , Séré Eric
Published: March 2016

In this paper we discuss the existence of normalized solutions of nonlinear elliptic PDEs in gaps of the essential spectrum of the corresponding differential operator. This issue can he of interest, for instance, in nonlinear optics or in crystalline models with impurities.
Some Notes on Function Spaces and Dirichlet Forms on Self-Similar Sets
Authors: Liu Yuan , Wen Zhi-Ying
Published: March 2016

Grigor’yan, Hu and Lau [10] introduced sub-Gaussian heat kernels on general metric measure spaces and defined a family of function spaces to characterize the domain of associated Dirichlet forms. In this paper, we will improve their results about norm equivalence. As an application, we construct self-similar Dirichlet forms on a class of self-similar sets containing the Sierpiński gaskets and carpets. Then we prove the Poincaré inequality and give effective resistance estimates by the self-similarity. Consequently, we have a new equivalent characterization of heat kernel estimates through function spaces with strong recurrent condition.
Homoclinic Orbits For First Order Hamiltonian Systems With Convex Potentials
Authors: Xu Xiangjin
Published: March 2016

In this paper new estimates on the C0-norm of homoclinic orbit are shown for first order convex Hamiltonian systems possessing super-quadratic potentials. Applying these estimates, some new results on the existence of infinitely many geometrically distinct homoclinic orbits are proved, which generalize the main results in [2] and [10].
Unique Strong Solutions and V -Attractors of a Three Dimensional System of Globally Modified Navier-Stokes Equations
Authors: Caraballo Tomás , Real José , Kloeden Peter E.
Published: March 2016

We prove the existence and uniqueness of strong solutions of a three dimensional system of globally modified Navier-Stokes equations. The flattening property is used to establish the existence of global V -attractors and a limiting argument is then used to obtain the existence of bounded entire weak solutions of the three dimensional Navier-Stokes equations with time independent forcing.
Study of the Singular Yamabe Problem in Some Bounded Domain of ℝn
Authors: Rébaï Yomna
Published: March 2016

In this paper, we extend the result of R. Mazzeo and F. Pacard in the following direction: Given Ω any bounded open regular subset of ℝn, n ≥ 3 and given x1, x2 ∈ Ω, we give a sufficient condition on x1, x2, for the problem

to have a positive weak solution in Ω with 0 boundary data, which is singular at each xi.
On Stable and Meta-stable Solutions of the Shadow System For the Geirer-Meinhardt Equation
Authors: Dancer E.N.
Published: March 2016

We show how the theory of finite Morse index solutions for small diffusion problems can be adapted to the shadow system for the Geirer-Meinhardt equation. In particular, we prove that the only possible stable solution of this shadow system is the one peak solution discussed by Wei and Ni, Takagi and Yanagida.
Symmetry Results For Solutions of a Semilinear Nonhomogeneous Problem
Authors: Mataloni Silvia
Published: March 2016

In this paper we show the axial symmetry of a particular type of solutions of the following semilinear nonhomogeneous problem

where B is the unit ball of ℝN, N ≥ 3,

u+ = 0 if u < 0 and φ1(x) > 0 is the eigenfunction of -Δ corresponding to the first eigenvalue λ1, with Dirichlet boundary conditions. The solutions considered exhibit one or two peaks and concentrate, as the parameter s goes to +∞.
Perturbation of Symmetry and Multiplicity of Solutions For Strongly Indefinite Elliptic Systems
Authors: Tarsi Cristina
Published: March 2016

We consider the following elliptic system:

where Ω ⊂ ℝN, N ≥ 3 is a smooth bounded domain. If h(x) ≡ k(x) ≡ 0, the system presents a natural ℤ2 symmetry, which guarantees the existence of infinitely many solutions. In this paper we show that the multiplicity structure can be maintained if (p,q) lies below a suitable curve in ℝ2.
Maslov-Type Index Theory For Symplectic Paths With Lagrangian Boundary Conditions
Authors: Liu Chun-gen
Published: March 2016

In this paper, we establish an index theory for symplectic paths starting from the identity with a Lagrangian boundary condition. We show that in some sense the index is the intersection number of a Lagrangian path generated by the symplectic path and a constant Lagrangian path. The various properties of this index theory are developed.
Existence of Infinitely Many Solutions For a Class of Semilinear Subelliptic Equations on Rational Carnot Groups
Authors: Loiudice Annunziata
Published: March 2016

We establish the existence of infinitely many solutions for the equation

-ΔGu + u = f(ξ,u), ξ ∈ G

where ΔG is a sublaplacian on a rational Carnot group G. The function f is assumed to be periodic with respect to a discrete co-compact subgroup of G and satisfy subcritical growth conditions.
Asymptotic Behavior of the Steklov Eigenvalues For the p−Laplace Operator
Authors: Pinasco Juan Pablo
Published: March 2016

In this paper we study the asymptotic behavior of the Steklov eigenvalues of the p- Laplacian. We show the existence of lower and upper bounds of a Weyl-type expansion of the function N(λ) which counts the number of eigenvalues less than or equal to λ, and we derive from them asymptotic bounds for the eigenvalues.
Existence and Regularity Results For Some Singular Elliptic Problems
Authors: Leonori Tommaso , Petitta Francesco
Published: March 2016

We are interested in the study of existence, uniqueness and regularity of solutions for nonlinear elliptic problems whose model is the linear boundary value problem

where 0 ∊ Ω. We will prove that such results strongly depend on the size of the lower order term, that is on the constant B.
Computations of Critical Groups and Applications to Nonlinear Differential Equations With Resonance
Authors: Guo Yuxia , Liu Jiaquan
Published: March 2016

In this paper, we develop a method to compute critical groups at degenerate critical points under more general conditions. The abstract results are used to study the existence and multiplicity of nontrivial solutions for nonlinear differential equations with resonance both at infinity and at zero.
Hypersurfaces of Prescribed k-Order Mean Curvature in Exterior Domains
Authors: Huang Yong , Xu Lu
Published: March 2016

In this paper, we prove some existence theorems for hypersurfaces of prescribed k-order mean curvature in exterior domains, which lead to the study of the existence of k-order mean curvature equation in exterior domains.
On a “Zero Mass” Nonlinear Schrödinger Equation
Authors: Azzollini Antonio , Pomponio Alessio
Published: March 2016

We look for positive solutions to the nonlinear Schrödinger equation

−Ɛ2∆u − V (x)f′(u) = 0

in ℝN, where V is a continuous bounded positive potential and f satisfies particular growth conditions which make our problem fall in the so called “zero mass case”. We prove an existence result for any Ɛ > 0, and a multiplicity result for Ɛ sufficiently small.
The Global Existence and Semiclassical Limit of Weak Solutions to Multidimensional Quantum Drift-diffusion Model
Authors: Chen Xiuqing
Published: March 2016

We establish the global weak solutions to quantum drift-diffusion model, a fourth order parabolic system, in two or there space dimensions with large initial value and periodic boundary conditions and furthermore obtain the semiclassical limit by entropy estimate and compactness argument.
Dynamics of a Contact Structure Along a Vector Field of its Kernel
Authors: Bahri Abbas , Xu Yongzhong
Published: March 2016

In this paper we prove that in order to define the homology of [3], the hypothesis that there exists a vector field in the kernel of the contact form which defines a dual form with the same orientation is not essential. The technique is quantitative: as we introduce a large amount of rotation near the zeroes of the vector field in the kernel, we track down the modification of the variational problem and provide bounds on a key quantity (denoted by τ).
On the Morse Index of a Functional Arising in Contact Form Geometry
Authors: Bahri Abbas , Xu Yongzhong
Published: March 2016

In this paper we study a functional at infinity associated to a contact form on a three dimensional manifold. The Morse index of this functional at infinity can be decomposed into two parts, one along the characteristic manifold and the other along the normal directions. We prove that we can redistribute the negative directions between the two subspaces through a local deformation of the contact form near a critical point at infinity.
Branching Points For a Class of Variational Equations Involving Potential With Parameter
Authors: Lu Fei
Published: March 2016

In this paper, under the framework of Ambrosetti [1], using mountain-pass theorem and topological degree theory, we extend the result of Ambrosetti [1] (in which H is independent of λ and C1) to the equation

Au + H(λ, u) = λu

with H jointly continuous in (λ, u) and H(λ, ·) of class C0, 1. We give a description of the structure of local bifurcation from isolated eigenvalues of A: the set of bifurcation solutions at each isolated eigenvalue of A contains at least one connected branch. Applications to the existence of branching points to semilinear elliptic partial differential equations are given.
An Elementary Proof of a Characterization of Constant Functions
Authors: De Marco Giuseppe , Mariconda Carlo , Solimini Sergio
Published: March 2016

We give two short and elementary proofs of a characterization of constants function by Brezis. Whereas the original proof involves some refined arguments on Sobolev spaces and BV functions, ours are based either on convolutions or on a sort of nonsmooth Mean Value Theorem which is new to our knowledge.
Compactness Property of a Singular Quasilinear Elliptic Equation
Authors: Chen Jianqing
Published: March 2016

We characterize a compactness property for a quasilinear equation with critical growth and singular term. Some applications of the compactness property are also pointed out.
Weak Solutions of the Problem ∆2u = un/n-4 with Prescribed Singular Sets
Authors: Rébaï Yomna
Published: March 2016

In this paper, we are interested in the following biharmonic equation:

with Navier boundary conditions on ∂Ω; where Ω is an open bounded domain of ℝn, with n ≥ 5, having a smooth boundary, and ∑ ⊂ Ω is either a set of finite number of points of Ω or a smooth closed submanifold of Ω without boundary. We construct positive solutions to this problem which are singular at ∑.
Positive Solutions of a System Arising from Angiogenesis
Authors: Delgado Manuel , Suárez Antonio
Published: March 2016

We study a system of equations arising from angiogenesis which contains a non- regular term that vanishes below a certain threshold. We are forced to modify the usual methods of bifurcation theory because of this loss of regularity. Nevertheless, we obtain results on the existence, uniqueness and permanence of a positive solution for the time-dependent problem; and the existence and uniqueness of a positive solution for the stationary one.
On the Global Analytic Integrability of the Belousov–Zhabotinskii System
Authors: Llibre Jaume , Valls Clàudia
Published: March 2016

The well-known Belousov-Zhabotinskii system can be written as

ẋ = s(x + y - qx2 - xy), ẏ = s-1(-y + fz - xy), ż = w(x - z)

with f, q, s, w ∈ ℝ and s ≠ 0. In this paper we characterize its global analytic first integrals.
Semiclassical Limit For the Nonlinear Klein Gordon Equation in Bounded Domains
Authors: Ghimenti Marco G. , Grisanti Carlo R.
Published: March 2016

We are interested in the existence of standing waves for the nonlinear Klein Gordon equation ε2□ψ + Wʹ(ψ) = 0 in a bounded domain D. A standing wave has the form ψ(t, x) = u(x)e-iwt/ε; for these solutions the Klein Gordon equation becomes

We want to use a Benci-Cerami type argument in order to prove a the existence of several standing waves localized in suitable points of D. The main result of this paper is that, under suitable growth condition on W, for ε suffciently small, we have at least cat(D) stationary solutions of equation (†), where cat(D) is the Ljusternik-Schnirelmann category. The proof is achieved by solving a constrained critical point problem via variational techniques.
Quasilinear Equations With Boundary Blow-up and Exponential Reaction
Authors: García-Melián Jorge
Published: March 2016

We consider the quasilinear elliptic problem Δpu = a(x)eu in a smooth bounded domain Ω of ℝN, with u = +∞ on ӘΩ. Here Δpu = div(|▽u|p-2▽u) is the p-Laplacian operator with p > 1, and a(x) is a continuous positive weight function which can be singular on ӘΩ. Our results include existence, uniqueness and exact boundary behavior of solutions.
On a Fourth Order Elliptic Problem with a Singular Nonlinearity
Authors: Cassani Daniele , Ó João Marcosy do , Ghoussoub Nassif
Published: March 2016

We study the following semilinear biharmonic equation:

, where 0 ≤ f ≤ 1 and BR ⊂ ℝN, N ≥ 1, is the ball centered in the origin of radius R. We prove, under Dirichlet boundary conditions u = Әu/Әƞ = 0 on ӘBR, the existence of λ* = λ*(R, f) > 0 such that for λ ∈ (0, λ*) there exists a minimal (classical) solution u̲λ < which satisfies 0 < u̲λ < 1. In the extremal case λ = λ*, we prove the existence of a weak solution which has finite energy and which is the unique solution even in a very weak sense. For λ > λ* there are no solutions of any kind. Estimates on λ*, stability properties of solutions and nonexistence results in the whole space are also established.
Uniqueness and Characterization of the Maximizers of Integral Functionals With Constraints
Authors: Draghici Cristina , Hajaiej Hichem
Published: March 2016

In this paper we maximize a class of functionals under certain constraints. We find necessary and sufficient conditions for these maximizers to exist and be unique. Moreover, we characterize them and discuss the optimality of our results by constructing counterexamples when one of the hypotheses does not hold.
Positive Solutions for the p-Laplacian and Bounds for its First Eigenvalue
Authors: Bueno H. , Ercole G. , Zumpano A.
Published: March 2016

We prove a result of existence and localization of positive solutions of the Dirich- let problem for -Δpu = w(x)f(u) in a bounded domain Ω, where Δp is the p-Laplacian, w is a weight function and the nonlinearity f(u) satisfies certain local bounds. As in previous results in radially symmetric domains by two of the authors, and in contrast with the hypotheses usually made, no asymptotic behavior is assumed on f. A positive solution is obtained by applying the Schauder Fixed Point Theorem and such approach allows us to construct ordered sub- and super-solutions for the problem, thus producing an iterative method to obtain a positive solution. Our result leads not only to asymptotic conditions on the nonlinearity which provides the existence of a sequence of positive solutions for the problem with arbitrarily large sup norm, but also to an estimate of the first eigenvalue λp(Ω, w) of the p-Laplacian operator with weight w. For w ≡ 1, we compare our lower bound for λp(Ω, 1) with that obtained by means of the Cheeger constant h(Ω). We give a characterization of this constant in terms of the solution of the torsional creep problem -Δpɸp = 1 in Ω with Dirichlet boundary data, which offers a good approximation of the first eigenvalue of the p-Laplacian for p near 1.
Solitary Waves for Quasilinear Schrödinger Equations Arising in Plasma Physics
Authors: Marcos do Ó João , Moameni Abbas
Published: March 2016

We study the quasilinear Schrödinger equation

izt = −∆z +W(x)z − η(|z|2)z − k[∆p(|z|2)]p′(|z|2)z in ℝ2,

where W : ℝ2 →ℝ is a positive potential and the nonlinearity η : ℝ2 × ℝ → ℝ has critical or sub-critical exponential growth. Quasilinear Schrödinger equations of this type have been studied as models of several physical phenomena such as superfluid film equation, in the theory of Heisenberg ferromagnets and magnons, in dissipative quantum mechanics and in condensed matter theory. In a suitable Orlicz space together with Trudinger-Moser inequality we establish an existence of standing wave solutions for this problem. The second order nonlinearity considered in this paper corresponds to the superfluid equation in plasma physics.
Existence of Solution for a Class of Quasilinear Systems
Authors: Alves Claudianor O. , Soares Sérgio H. M.
Published: March 2016

This paper proves the existence of nontrivial solution for a class of quasilinear systems on bounded domains in ℝN, N ≥ 2, whose nonlinearity has a double criticality. The proof is based on a linking theorem without the Palais-Smale condition.
Nodal Solutions for a Singularly Perturbed Nonlinear Elliptic Problem on Riemannian Manifolds
Authors: Micheletti Anna Maria , Pistoia Angela
Published: March 2016

Let (M, g) be a smooth compact Riemannian N−manifold, N ≥ 2. We show that if the scalar curvature of g is not constant, the problem

−ε2∆gu + u = up−1 in M

has a positive solution with two positive peaks ξ1
ε and ξ2
ε, and a sign changing solution with one positive peak η1
ε and one negative peak η2
ε, such that as ε goes to zero Scalg(ξ1
ε), Scalg(η1
ε) →

Scalg(ξ) and Scalg(ξ2
ε), Scalg(η2
ε) →

Scalg(ξ). Here p > 2 if N = 2 and
Radial Solutions for Hamiltonian Elliptic Systems With Weights
Authors: De Nápoli Pablo L. , Drelichman Irene , Durán Ricardo G.
Published: March 2016

We prove the existence of infinitely many radial solutions for elliptic systems in ℝn with power weights. A key tool for the proof will be a weighted imbedding theorem for fractional-order Sobolev spaces, that could be of independent interest.
On the Poincaré-Hopf Theorem for Functionals Defined on Banach Spaces
Authors: Cingolani Silvia , Degiovanni Marco
Published: March 2016

Let X be a reflexive Banach space and f : X → ℝ a Gâteaux differentiable function with f′ demicontinuous and locally of class (S)+. We prove that each isolated critical point of f has critical groups of finite type and that the Poincaré-Hopf formula holds. We also show that quasilinear elliptic equations at critical growth are covered by this result.
On the Multiplicity of Orthogonal Geodesics in Riemannian Manifold With Concave Boundary. Applications to Brake Orbits and Homoclinics
Authors: Giambò Roberto , Giannoni Fabio , Piccione Paolo
Published: March 2016

Let (M, g) be a complete Riemannian manifold, Ω ⊂ M an open subset whose closure is diffeomorphic to an annulus. If ∂Ω is smooth and it satisfies a strong concavity assumption, then it is possible to prove that there are at least two geometrically distinct geodesics in Ω̅ = Ω ∪ ∂Ω starting orthogonally to one connected component of ∂Ω and arriving orthogonally onto the other one. The results given in [6] allow to obtain a proof of the existence of two distinct homoclinic orbits for an autonomous Lagrangian system emanating from a nondegenerate maximum point of the potential energy, and a proof of the existence of two distinct brake orbits for a class of Hamiltonian systems. Under a further symmetry assumption, it is possible to show the existence of at least dim(M) pairs of geometrically distinct geodesics as above, brake orbits and homoclinics.
Invariant Manifolds for Random and Stochastic Partial Differential Equations
Authors: Caraballo Tomás , Duany Jinqiao , Lu Kening , Schmalfuβ Björn
Published: March 2016

Random invariant manifolds are geometric objects useful for understanding complex dynamics under stochastic infuences. Under a nonuniform hyperbolicity or a nonuniform exponential dichotomy condition, the existence of random pseudo- stable and pseudo-unstable manifolds for a class of random partial differential equations and stochastic partial differential equations is shown. Unlike the in- variant manifold theory for stochastic ordinary differential equations, random norms are not used. The result is then applied to a nonlinear stochastic partial differential equation with linear multiplicative noise.
Pseudo Focal Points Along Lorentzian Geodesics and Morse Index
Authors: Javaloyes Miguel Ángel , Masiello Antonio , Piccione Paolo
Published: March 2016

Given a Lorentzian manifold (M, g), a geodesic γ in M and a timelike Jacobi field γ along γ, we introduce a special class of instants along γ that we call γ- pseudo conjugate (or focal relatively to some initial orthogonal submanifold). We prove that the γ-pseudo conjugate instants form a finite set, and their number equals the Morse index of (a suitable restriction of) the index form. This gives a Riemannian-like Morse index theorem. As special cases of the theory, we will consider geodesics in stationary and static Lorentzian manifolds, where the Jacobi field γ is obtained as the restriction of a globally defined timelike Killing vector field.
Nonuniform Stability of Difference Equations with Infinite Delay
Authors: Barreira Luis , Valls Claudia
Published: March 2016

In this note we consider the notion of nonuniform exponential contraction for delay difference equations with infinite delay (one can argue that the only delay difference equations are those with infinite delay, since otherwise we can always bring them to the standard recurrence form in some higher-dimensional space). We consider the general case of a nonautonomous dynamics. Our main objective is to show that exponential contractions persist under sufficiently small linear and nonlinear perturbations. This includes establishing the continuous dependence with the perturbation of the constants in the notion of contraction. We also characterize the nonuniform exponential contractions in terms of strict Lyapunov sequences, in particular by constructing explicitly a strict Lyapunov sequence for each exponential contraction.
On a Q–Curvature Equation on Complete Riemannian Manifolds
Authors: Maliki Youssef
Published: March 2016

We proceed by variational method to prove the existence of solution of a fourth order partial differential equation on complete non compact manifolds. Sufficient conditions, including the geometry of the manifold, are given to make a minimizing sequence converge to a non trivial solution belonging to Cloc
4,α (M) for some α ∈ (0, 1).
On Ground State Solutions to Mixed Type Singular Semi-Linear Elliptic Equations
Authors: Mohammed Ahmed
Published: March 2016

The purpose of the paper is to establish the existence of ground state solutions to -Δu = ηa(x)f(u)+γb(x)g(u), where the locally Hölder continuous non-negative functions a and b and the non-linearities f and g satisfy some general conditions.
Stability of Closed Characteristics on Compact Hypersurfaces in R2n Under a Pinching Condition
Authors: Wang Wei
Published: March 2016

In this article, let ∑ ⊂R2n be a compact convex hypersurface which is (r,R)-pinched with

. Then ∑ carries at least two strictly elliptic closed characteristics; moreover, ∑ carries at least

non-hyperbolic closed characteristics.
Bifurcation and Stability of Travelling Waves in Self-focusing Planar Waveguides
Authors: Genoud François
Published: March 2016

A mathematical problem arising in the modelling of travelling waves in self- focusing waveguides is studied. It involves a nonlinear Schrödinger equation in ℝ1+1. In the paraxial approximation, the travelling waves are interpreted as standing waves of the NLS. Local and global bifurcation results are established for the corresponding \stationary" equation. The combination of variational arguments with an implicit function theorem yields smooth branches of positive solutions parametrized by the \frequency" of the standing wave of NLS. Precise informations on the asymptotic behaviour of various norms of the solutions along the branches are given. In the case of even symmetry, a global bifurcation result is proved, providing a branch of positive even solutions parametrized by frequencies in the whole half-line (0, ∞). The stability of the standing waves of the NLS relies on the monotonicity of their L2-norm as a function of the frequency. This criterion, precisely going back to formal arguments in the physical literature on self-focusing waveguides, is proved to hold along the whole global branch of solutions in the even case. The mathematical results are applied to the study of a non-homogeneous planar self-focusing waveguide with a Kerr-type nonlinear response, yielding existence of TE travelling waves and their stability among the set of TE modes. The bifurcation analysis gives informations on the power of the beam. In particular, it provides the possibility of low power cut-off.
Multiplicity Results for a Class of Asymptotically Linear Elliptic Problems With Resonance and Applications to Problems With Measure Data
Authors: Ferrero Alberto , Saccon Claudio
Published: March 2016

We study existence and multiplicity results for solutions of elliptic problems of the type -Δu = g(x; u) in a bounded domain Ω with Dirichlet boundary conditions. The function g(x; s) is asymptotically linear as |s| → +∞. Also resonant situations are allowed. We also prove some perturbation results for Dirichlet problems of the type -Δu = gƐ(x, u) where gε(x, s) → g(x, s) as ε → 0. The previous results find an application in the study of Dirichlet problems of the type -Δu = g(x, u)+μ where μ is a Radon measure. To properly set the above mentioned problems in a variational framework we also study existence and properties of critical points of a class of abstract nonsmooth functional defined on Banach spaces and extend to this nonsmooth framework some classical linking theorems.
Existence of Multiple Solutions for Nonlinear Dirichlet Problems with a Nonhomogeneous Differential Operator
Authors: Kyritsi Sophia Th. , O’ Regan Donal , Papageorgiou Nikolaos S.
Published: March 2016

We consider nonlinear elliptic problems driven by a nonhomogeneous nonlinear differential operator. Using variational methods combined with Morse theory (critical groups), we prove two multiplicity results establishing three nontrivial smooth solutions. For the semilinear problem (linear differential operator), we produce four nontrivial smooth solutions. In the special case of the p-Laplacian differential operator, our framework of analysis incorporates equations which are resonant at infinity with respect to the principal eigenvalue.
Complete Blow-up and Avalanche Formation for a Parabolic System with Non-Simultaneous Blow-up
Authors: Brändle Cristina , Quirós Fernando , Rossi Julio D.
Published: March 2016

We study the possibility of defining a nontrivial continuation after the blow-up time for a system of two heat equations with a nonlinear coupling at the boundary. It turns out that any possible continuation that verifies a maximum principle is identically infinity everywhere after the blow-up time; that is, both components blow up completely. We also analyze the propagation of the singularity to the whole space, the avalanche, when blow-up is non-simultaneous.
Orbital Stability Property for Coupled Nonlinear Schrödinger Equations
Authors: Maia Liliane de Almeida , Montefusco Eugenio , Pellacci Benedetta
Published: March 2016

Orbital stability property for weakly coupled nonlinear Schrödinger equations is investigated. Different families of orbitally stable standing waves solutions will be found, generated by different classes of solutions of the associated elliptic problem. In particular, orbitally stable standing waves can be generated by least action solutions, but also by solutions with one trivial component whether or not they are ground states. Moreover, standing waves with components propagating with the same frequencies are orbitally stable if generated by vector solutions of a suitable Schrödinger weakly coupled system, even if they are not ground states.
The Wente Problem Associated to the Modified Helmholtz Operator on Weighted Sobolev Spaces
Authors: Omrane Ines Ben , Jleli Mohamed , Samet Bessem
Published: March 2016

In this paper, we give a weighted version of regularity of solutions of the Wente problem associated to the modified Helmholtz operator -Δ + αI, where α is a positive constant.
On the Decay in Time of Solutions of the Generalized Regularized Boussinesq System
Authors: Mammeri Youcef
Published: March 2016

We are interested in dispersive properties of the Boussinesq system for small initial data. We prove that, for a suficiently high order nonlinearity, the solution of the Boussinesq system exists for all time and tends to zero as the time tends to infinity.
On the Symmetry of the Ground States of Nonlinear Schrödinger Equation with Potential
Authors: Maeda Masaya
Published: March 2016

We investigate the minimizers of the energy functional

under the constraint of the L2-norm. We show that for the case L2-norm is small, the minimizer is unique and for the case L2-norm is large, the minimizer concentrate at the maximum point of b and decays exponentially. By this result, we can show that if V and b are radially symmetric but b does not attain its maxi- mum at the origin, then the symmetry breaking occurs as the L2-norm increases. Further, we show that for the case b has several maximum points, the minimizer concentrates at a point which minimizes a function which is defined by b, V and the unique positive radial solution of -∆φ + φ - φp = 0. For the case when V and b are radially symmetric, we show that if the minimizer concentrates at the origin, then the minimizer is radially symmetric. Further, we construct an energy functional such that the minimizer breaks its symmetry once but after that it recovers to be symmetric as the L2-norm increases.
Functions on Tori with Minimal Number of Critical Points and Rotation Type Solutions of Spatially Periodic Hamiltonian Systems
Authors: Qiao Hui , Long Yiming
Published: March 2016

In this paper we use functions on Tn−1 with n ≥ 2 possessing minimal number of critical points given by Ljusternik-Schnirelman theory to construct spatially n-periodic Hamiltonian systems which possess precisely n distinct rotation type solutions with prescribed energy and rotation type vector.
Symmetric Joins and Weighted Barycenters
Authors: Kallel Sadok , Karoui Rym
Published: March 2016

Given a space X, we study the homotopy type of Bn(X) the space obtained as the “union of all (n−1)-simplexes spanned by points in X” or the space of “formal barycenters of weight n or less” of X. This is a space encountered in non-linear analysis under the name of space of barycenters or in differential geometry in the case n = 2 as the space of chords. We first relate this space to a more familiar symmetric join construction and then determine its stable homotopy type in terms of the symmetric products on the suspension of X. This leads to a complete understanding of the homology of Bn(X) as a functor of X, and to an expression for its Euler characteristic given in terms of X. A sharp connectivity theorem is also established. Finally the case of spheres S is studied in details and the homotopy type of Bn(S ) is described, generalizing in this way an early and beautiful result of James, Thomas, Toda and Whitehead.
Elliptic-Parabolic Equation with Absorption of Obstacle type
Authors: Igbida Noureddine , Karami Fahd
Published: March 2016

This paper is concerned with existence and uniqueness of solutions for a doubly nonlinear degenerate parabolic problem of the type β(w)t − div a(x, Dw) + ∂ j ( ., β(w) ) ∋ f, where α is a Leray-Lions operator, β is a nondecreasing continuous function and ∂ j(., r) is a maximal monotone graph with respect to r defined on a closed interval of ℝ. Particular cases of j correspond to the so called obstacle problem.
On Radial Solutions of the Schrödinger Type Equation
Authors: Chabrowski Jan H. , Grotowski Joseph F.
Published: March 2016

We establish compact embeddings of the radial Sobolev space Hrad
1,p(ℝN) into weighted Lebesgue spaces Lq
w (ℝN) under various assumptions on the weight function w. We use these, along with a variety of variational techniques, to prove the existence of nontrivial nonnegative solutions for a class of nonlinear Schrödinger type equations.
Limit Cycles Bifurcating from a 2-Dimensional Isochronous Torus in ℝ3
Authors: Llibre Jaume , Rebollo-Perdomo Salomón , Torregrosa Joan
Published: March 2016

In this paper we illustrate the explicit implementation of a method for computing limit cycles which bifurcate from a 2-dimensional isochronous set contained in ℝ3, when we perturb it inside a class of differential systems. This method is based in the averaging theory. As far as we know all applications of this method have been made perturbing noncompact surfaces, as for instance a plane or a cylinder in ℝ3. Here we consider polynomial perturbations of degree d of an isochronous torus. We prove that, up to first order in the perturbation, at most 2(d+1) limit cycles can bifurcate from a such torus and that there exist polynomial perturbations of degree d of the torus such that exactly ν limit cycles bifurcate from such a torus for every ν ∈ {2, 4,...,2(d + 1)}.
Homogenization and Vanishing Viscosity in Fully Nonlinear Elliptic Equations: Rate of Convergence Estimates
Authors: Camilli Fabio , Cesaroni Annalisa , Marchi Claudio
Published: March 2016

This paper is devoted to studying the behavior as ε → 0 of the equations

uε + H(x, x/ε, Duε, εγD2uε) = 0

with γ > 0. It is known that, under some periodicity and ellipticity or coercivity assumptions, the solution uε converges to the solution u of an effective equation u + H̅(x, Du) = 0, with an effective Hamiltonian H̅ dependent on the value of γ. The main purpose of this paper is to estimate the rate of convergence of uε to u. Moreover we discuss some examples and model problems.
Smooth Dependence of Topological Conjugacies on Parameters
Authors: Barreira Luis , Valls Claudia
Published: March 2016

For nonautonomous linear equations xʹ = A(t)x with a nonuniform exponential dichotomy, we show that under sufficiently small C1 parameter-dependent perturbations, namely of the form xʹ = A(t)x + f (t, x, λ), the topological conjugacies in an appropriate version of the Grobman-Hartman theorem are of class C1 in the parameter. The proof uses the fiber contraction principle.
Quasilinear Riccati Type Equations and Quasiminimizers
Authors: Martio O.
Published: March 2016

Solutions u of the equation -∇ ・ A(x, ∇u) = b(x)|∇u|q are studied in the subcritical case p - 1 ≤ q ≤ p/n + p - 1, q < p < n. Here A(x, h) satisfies A(x, h) ・ h ≈ |h|p and b a bounded function in an open set Ω ⊂ Rn. It is shown that the solutions u are local quasiminimizers in Ω. From the quasiminimizing property several conclusions on the behavior of the solutions u can be derived. In particular, the method provides an easy solution to the uniqueness problem in the class W0
1,p(Ω).
Almost Periodic Solutions of Monotone Second-Order Differential Equations
Authors: Ayachi Moez , Blot Joël , Cieutat Philippe
Published: March 2016

We give sufficient conditions for the existence of almost periodic solutions of the secondorder differential equation

u′′(t) = f (u(t)) + e(t)

on a Hilbert space H, where the vector field f : H → H is monotone, continuous and the forcing term e : ℝ → H is almost periodic. Notably, we state a result of existence and uniqueness of the Besicovitch almost periodic solution, then we approximate this solution by a sequence of Bohr almost periodic solutions.
The Sturm-Liouville Hierarchy of Evolution Equations
Authors: Johnson Russell , Zampogni Luca
Published: March 2016

We introduce a hierarchy of evolution equations based on the Sturm-Liouville equation −(pφ′)′ + qφ = λyφ. Our hierarchy includes the Korteweg-de Vries (K-dV) and the Camassa-Holm (CH) hierarchy. We determine a class of solutions of the hierarchy which are of algebro-geometric type. The initial condition of such a solution is drawn from a finite-gap isospectral class of the Sturm-Liouville equation.
Dynamics of Periodic Second-Order Equations Between an Ordered Pair of Lower and Upper Solutions
Authors: Ureña Antonio J.
Published: March 2016

We consider periodic second-order equations having an ordered pair of lower and upper solutions and show the existence of asymptotic trajectories heading towards the maximal and minimal periodic solutions which lie between them.
Multiple Solutions for Nonlinear Neumann Problems with Asymmetric Reaction, via Morse Theory
Authors: Gasiński Leszek , Papageorgiou Nikolaos S.
Published: March 2016

We consider a nonlinear Neumann problem driven by the p-Laplacian and with a reaction which exhibits an asymmetric behaviour near +∞ and near −∞. Namely, it is (p − 1)- superlinear near +∞ (but need not satisfy the Ambrosetti-Rabinowitz condition) and it is (p − 1)-linear near −∞. Combining variational methods with Morse theory, we show that the problem has at least three nontrivial smooth solutions.
Odd Homoclinic Orbits for a Second Order Hamiltonian System
Authors: Maia Liliane A. , Miyagaki Olimpio H. , Soares Sergio H. M.
Published: March 2016

The aim of this paper is to find an odd homoclinic orbit for a class of reversible Hamiltonian systems. The proof is variational and it employs a version of the concentration compactness principle of P. L. Lions in a lemma due to Struwe.
On the Cheeger–Buser Type Inequalities for the p-Laplacian
Authors: Matei Ana-Maria
Published: March 2016

We study the relationship between the first eigenvalue of the p-Laplacian (p > 1) and Cheeger’s isoperimetric constant for families of compact manifolds and graphs with the Cheeger constant converging to zero.
Symmetrization for Neumann Anisotropic Problems and Related Questions
Authors: Pietra Francesco Della , Gavitone Nunzia
Published: March 2016

In this paper we prove some comparison results for Neumann elliptic problems whose model involves the anisotropic Laplacian

ΔHu = div (H(Du)Hξ(Du)),

where H is a positively homogenous convex function. Finally, we find a Poincaré inequality in the anisotropic setting.
Nonlinear Schrödinger Equations with Sign-Changing Potential
Authors: He Haiyang
Published: March 2016

In this paper, we show that the following system

−Δu + λV(x)u = g(x, v), −Δv + λV(x)v = f (x, u), x ∈ ℝN (0.1)

possesses at least one non-trivial solution pair (u, v) for λ > 0 large enough, where f (x, t), g(x, t) are continuous functions on ℝN × ℝ and super-linear at t = 0 as well as at t = +∞, V(x) is allowed to be sign-changing. We mention that we do not assume that f or g satisfies the Ambrosetti-Rabinowitz condition.