bluuebunny/crossref_metadata_2025_split huggingface dataset:
Bifurcation and Stability of Travelling Waves in Self-focusing Planar Waveguides
Authors: Genoud François 
Published: March 2016

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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

Abstract
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.
Multiple Solutions for Nonlinear Neumann Problems with Asymmetric Reaction, via Morse Theory
Authors: Gasiński Leszek , Papageorgiou Nikolaos S. 
Published: March 2016

Abstract
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

Abstract
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.
Symmetrization for Neumann Anisotropic Problems and Related Questions
Authors: Pietra Francesco Della , Gavitone Nunzia 
Published: March 2016

Abstract
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

Abstract
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.
Steady States, Global Existence and Blow-up for Fourth-Order Semilinear Parabolic Equations of Cahn–Hilliard Type
Authors: Álvarez-Caudevilla Pablo , Galaktionov Victor A. 
Published: March 2016

Abstract
Fourth-order semilinear parabolic equations of the Cahn-Hilliard-type
ut + Δ2u = γu ± Δ(|u|p−1u) in Ω × ℝ+, (0.1)
are considered in a smooth bounded domain Ω ⊂ ℝN with Navier-type boundary conditions on ∂Ω, or Ω = ℝN, where p > 1 and γ are given real parameters. The sign “+” in the “diffusion term” on the right-hand side means the stable case, while “−” reflects the unstable (blow-up) one, with the simplest, so called limit, canonical model for γ = 0,
ut + Δ2u = ±Δ(|u|p−1u) in ℝN × R+. (0.2)
The following three main problems are studied:
(i) for the unstable model (0.1), with the −Δ(|u|p−1u), existence and multiplicity of classic steady states in Ω ⊂ ℝN and their global behaviour for large γ > 0;
(ii) for the stable model (0.2), global existence of smooth solutions u(x, t) in ℝN × R+ for bounded initial data u0(x) in the subcritical case

; and
(iii) for the unstable model (0.2), a relation between finite time blow-up and structure of regular and singular steady states in the supercritical range. In particular, three distinct families of Type I and II blow-up patterns are introduced in the unstable case.
Persistence Properties of the Two-Component b-Family System
Authors: Zhu Min , Xu Junxiang 
Published: March 2016

Abstract
In this paper we study the persistence properties of decay for the solutions to the two component b-family system. Using the method of characteristics, we establish that certain decay properties of the initial data persist as long as the solution exists. We also examine the propagation behavior of compactly supported solutions. We show that solutions have an infinite speed of propagation, that is, a non-trivial strong solution with compact initial value can not be compactly supported at any later time.
Multiple Sign Changing Solutions of Nonlinear Elliptic Problems in Exterior Domains
Authors: Clapp Mónica , Salazar Dora 
Published: March 2016

Abstract
We consider the problem
−Δu + (V∞ + V(x))u = |u|p−2 u, u ∈ H0
1 (Ω),
where Ω is an exterior domain in ℝN, V∞ > 0, V ∈ C0(ℝN), infℝN V > −V∞ and V(x) → 0 as |x| → ∞. Under symmetry conditions on Ω and V, and some assumptions on the decay of V at infinity, we show that there is an effect of the topology of the orbit space of certain subsets of the domain on the number of low energy sign changing solutions to this problem.
A Note on Vortices with Prescribed Charge
Authors: Badiale Marino , Rolando Sergio 
Published: March 2016

AbstractWe consider the nonlinear Klein-Gordon equation with nonnegative potential, which makes the equation suitable for physical models, and prove the existence of solitary wave solutions with nonvanishing angular momentum and enough largely prescribed charge (c-vortices). This is done by solving the following minimization problem:where x = (y, z) ∈ ℝ2× ℝ, k ≠ 0 and H is a suitable subspace of H1(ℝ3) × ℝ.
Nonlinear Elliptic Equations in Strip-Like Domains
Authors: Byeon Jaeyoung , Tanaka Kazunaga 
Published: March 2016

Abstract
We study the existence of a positive solution of a nonlinear elliptic equation

where k ≥ 2 and D is a bounded domain domain in Rℓ, ℓ ≥ 1. We give almost necessary and sufficient condition of f(ξ) for the existence of a positive solution, which is inspired by the works of Berestycki-Lions [1] and Berestycki-Gallouët-Kavian [3].
Beyond the Mountain Pass: Some Applications
Authors: Peral Ireneo 
Published: March 2016

Abstract
We will survey three applications of the famous Mountain Pass Lemma by Antonio Ambrosetti and Paul H. Rabinowitz, [6]. Precisely, we will present some results on the following problems:
A model of growth and roughening of surfaces, usually called Kardar-Parisi-Zhang model.
A supercritical problem involving Hardy-Leray potential with the pole at the boundary.
A fourth order nonlinear problem related to epitaxial growth.
The common characteristic of all these problems is that we have some extra difficulties in order to apply the Mountain Pass Lemma.
On the Uniformity of the Constant in the Poincaré Inequality
Authors: Ruiz David 
Published: March 2016

Abstract
The classical Poincaré inequality establishes that for any bounded regular domain Ω ⊂ ℝN there exists a constant C = C(Ω) > 0 such that

In this paper we show that C can be taken independently of Ω when Ω is in a certain class of domains. Our result generalizes previous results in this direction.
Elliptic Variational Inequalities with Discontinuous Multi-Valued Lower Order Terms
Authors: Carl Siegfried 
Published: March 2016

AbstractWe consider multi-valued elliptic variational inequalities for operators of the formu ↦ Au + ∂2ψ(u, u),where A is a second order elliptic operator of Leray-Lions type, and u ↦ ∂2ψ(u, u) is a multi-valued lower order term that may neither be lower nor upper semicontinuous, see e.g., Figure 2. More precisely, the lower order term is generated by some function (r, s) → ψ(r, s) with ψ : ℝ × ℝ → ℝ that is locally Lipschitz with respect to its second variable s for each r ∈ ℝ, and ∂2ψ(r, s) denotes Clarke’s generalized gradient of s ↦ ψ(r, s). The novelty of this paper is that the multifunction r ↦ ∂2ψ(r, s) may discontinuously depend on r in a certain specified way, which gives rise to the new class of discontinuous multi-valued lower order terms s ↦ ∂2ψ(s, s). Though rich in structure, the characteristic features of this new class of multi-valued lower order terms can easily be described and verified. Our main goal is to provide an analytical framework and to prove existence and comparison results for this new class of multi-valued variational inequalities that includes the theory of variational-hemivariational inequalities as special case.
Optimization of the First Eigenvalue of Equations with Indefinite Weights
Authors: Cosner Chris , Cuccu Fabrizio , Porru Giovanni 
Published: March 2016

Abstract
We investigate minimization and maximization of the principal eigenvalue of the Laplacian under Dirichlet boundary conditions in case the weight has indefinite sign and varies in a class of rearrangements. Biologically, such optimization problems are motivated by the question of determining the most convenient spatial arrangement of favorable and unfavorable resources for a species to survive or to decline. The question may have practical importance in the context of reserve design or pest control.
Quasilinear Elliptic Equations on Half- and Quarter-spaces
Authors: Dancer E.N. , Du Yihong , Efendiev Messoud 
Published: March 2016

Abstract
We consider quasilinear elliptic problems of the form Δpu + f(u) = 0 over the half-space H = {x ∈ ℝN : x1 > 0} and over the quarter-space Q = {x ∈ ℝN : x1 > 0, xN > 0}. In the half-space case we assume u ≥ 0 on ∂H, and in the quarter-space case we assume that u ≥ 0 on {x1 = 0} and u = 0 on {xN = 0}. Let u ≢ 0 be a bounded nonnegative solution. For some general classes of nonlinearities f , we show that, in the half-space case, limx1→∞ u(x1, x2, ..., xN) always exists and is a positive zero of f ; and in the quarter-space case,

where V is a solution of the one-dimensional problem
ΔpV + f(V) = 0 in ℝ+, V(0) = 0, V(t) > 0 for t > 0, V(+∞) = z,
with z a positive zero of f . Our results extend most of those in the recent paper of Efendiev and Hamel [6] for the special case p = 2 to the general case p > 1. Moreover, by making use of a sharper Liouville type theorem, some of the results in [6] are improved. To overcome the difficulty of the lack of a strong comparison principle for p-Laplacian problems, we employ a weak sweeping principle.
Periodic Bouncing Solutions for Nonlinear Impact Oscillators
Authors: Fonda Alessandro , Sfecci Andrea 
Published: March 2016

Abstract
We prove the existence of a periodic solution to a nonlinear impact oscillator, whose restoring force has an asymptotically linear behavior. To this aim, after regularizing the problem, we use phase-plane analysis, and apply the Poincaré-Bohl fixed point Theorem to the associated Poincaré map, so to find a periodic solution of the regularized problem. Passing to the limit, we eventually find the “bouncing solution” we are looking for.
Positive, Negative, and Nodal Solutions to Elliptic Differential Inclusions Depending on a Parameter
Authors: Iannizzotto Antonio , Marano Salvatore A. , Motreanu Dumitru 
Published: March 2016

Abstract
The homogeneous Dirichlet problem for a partial differential inclusion involving the p- Laplace operator and depending on a parameter λ > 0 is investigated. The existence of three smooth solutions, a smallest positive, a biggest negative, and a nodal one, is obtained for any λ sufficiently large by combining variational methods with truncation techniques.