bluuebunny/crossref_metadata_2025 huggingface dataset:
Solitary Waves with Non-Vanishing Angular Momentum
Authors: Benci Vieri , Visciglia Nicola
Published: March 2016

In this paper we prove the existence of infinitely many solitary waves with non vanishing angular momentum for Nonlinear Schrödinger Equation:

where ψ, : ℝ3 × ℝ → ℂ and 2 < p < 6.
On Existence and Concentration of Solutions for a Class of Hamiltonian Systems in ℝN
Authors: Alves C.O. , Soares S.H.M. , Yangt Jianfu
Published: March 2016

In this paper we are concerned with the existence of solutions for the Hamiltonian system

where N ≥ 3, Q, K : ℝN → ℝ are bounded positive continuous functions and the numbers p, q > 1 satisfy

Using dual variational methods we show that under certain conditions on Q and K the system (S∊) has a positive solution which concentrates as ∊ → 0 in a point xo ∈ ℝN where related functionals realize its minimum energy.
Asymptotic Estimates of the First Eigenvalue of the p-Laplacian
Authors: Colbois Bruno , Matei Ana-Maria
Published: March 2016

We consider a 1-parameter family of hyperbolic surfaces M(t) of genus ν which degenerate as t → 0 and we obtain a precise estimate of λ1,p(t), the first eigenvalue of the p-Laplacian (p > 1) on M(t). In some cases we also give a precise estimate of the first eigenfunctions.

As a direct application, we obtain that the quotient

(which is invariant under scaling of the metric) is unbounded even on the set of Riemannian manifolds with constant sectional curvature. This is to our knowledge, the first example of a family of manifolds with this property.

To prove our results we use in an essential way the geometry of hyperbolic surfaces which is very well known. We show that an eigenfunction for λ1,p(t) of Lp norm one is almost constant in the Lp sense (as t → 0) on the parts of M(t) with large injectivity radius, and we estimate precisely its p-energy on the parts with small injectivity radius.
Existence and Nonexistence of Positive Solutions of Indefinite Elliptic Problems in ℝN
Authors: Schneider Matthias
Published: March 2016

Our purpose is to find positive solutions u ∈ D1,2 (ℝN) of the semilinear elliptic problem -Δu - λV(x)u = h(x)up-1 for 2 < p. The functions V and h may have an indefinite sign and the linearized operator need not have a fist (principal) eigenvalue, e.g. we allow V ≡ 1. We give precise existence and nonexistence criteria, which depend on λ and on the growth of h- and h+/V+. Existence theorems are obtained by constrained minimization. The mountain pass theorem leads to a second solution.
Varying Stoichiometric Exponents I: Classical Steady-States and Metasolutions
Authors: López-Gómez Julián
Published: March 2016

This paper characterizes the existence of classical solutions for a class of semilinear elliptic boundary value problems of logistic type whose stoichiometric coefficient is allowed to vary within the region where the reaction takes place. Most of the effort will be focused on the problem of the existence of metasolutions; large solutions extended by infinity that are the main candidates for describing the dynamics of the model in the absence of classical steady-states.
Geodesics in Static Lorentzian Manifolds with Critical Quadratic Behavior
Authors: Bartolo R. , Candela A.M. , Flores J.L. , Sánchez M.
Published: March 2016

The aim of this paper is t o study the geodesic connectedness of a complete static Lorentzian manifold (M.〈·, ·〉L) such that. if K is its irrotational timelike Killing vector field, the growth of β = -〈K, K〉L is (at most) quadratic. This is shown to be equivalent to a critical variational problem on a Riemannian manifold, where some geometrical interpretations yield the optimal results. Multiplicity of connecting geodesics is studied, especially in the timelike case, where the restrictions for the number of causal connecting geodesics are stressed. Extensions to the non-complete case, including discussions on pseudoconvexity, are also given.
On the Quadratic Endomorphisms of the Plane
Authors: Bofill F. , Garrido J.L. , Vilamajó F. , Romero N. , Rovella A.
Published: March 2016

In this article we establish the following result: if a nondegenerate quadratic endomorphism of the plane has no fixed points, then every point has empty omega-limit set and alpha-limit set. It is also shown that there exists a six parameter family open and dense in the space of all quadratic mappings of the plane (even those having fixed points). The degenerate case (when the quadratic forms of both components are linearly dependent), for which the theorem fails, is considered in the last section.
A Seven-Positive-Solutions Theorem for a Superlinear Problem
Authors: Gaudenzi Marcellino , Habets Patrick , Zanolin Fabio
Published: March 2016

We consider the superlinear boundary value problem

uʺ + aμ(t)uγ+1 = 0, u(0) = 0, u(1) = 0,

where γ > 0 and aμ(t) is a sign indefinite weight of the form a+(t)−μa−(t). We prove, for μ positive and large, the existence of 2k − 1 positive solutions where k is the number of positive humps of aμ(t) which are separated by k − 1 negative humps. For sake of simplicity, the proof is carried on for the case k = 3 yielding to 7 positive solutions. Our main argument combines a modified shooting method in the phase plane with some properties of the blow up solutions in the intervals where the weight function is negative.
Combining Linear and Nonlinear Diffusion
Authors: Delgado M. , López-Gómez J. , Suárez A.
Published: March 2016

In this paper we study a generalized porous medium equation where the diffusion rate, say m(x) -spatially heterogeneous- is assumed to be linear, m = 1, on a piece of the support domain, Ω1 and slow nonlinear, m(x) > 1, in its complement, Ωm := Ω \ Ω̄1. More precisely, we characterize the existence of positive solutions and construct the corresponding global bifurcation diagram as one of the parameters of the model changes, showing that a continuous transition occurs between the diagrams of the completely linear case (Ω = Ω1) and of the completely nonlinear case (­Ωm = Ω­). As a result, the effect of a localized slow diffusion rate with varying support is completely characterized. Our analysis is imperative in order to design porous media multi-components systems with changing diffusion rates.
Non-Existence Results for the Coupled Klein-Gordon-Maxwell Equations
Authors: D’Aprile Teresa , Mugnai Dimitri
Published: March 2016

In this paper we obtain some non-existence results for the Klein-Gordon equation coupled with the electrostatic field. The method relies on the deduction of some suitable Pohožaev identity which provides necessary conditions to get existence of nontrivial solutions. The case of Maxwell-Schrödinger type coupled equations is also considered.
A Simple Approach to Brouwer Degree Based on Differential Forms
Authors: Mawhin Jean
Published: March 2016

We present Heinz’ approach to Brouwer degree in a simpler, shorter and better motivated way. We link it to Kronecker index, use the language of differential forms at an elementary level, and prove the homotopy invariance using an unpublished result of Tartar.
On the Ambrosetti-Rabinowitz Superlinear Condition
Authors: Liu Zhaoli , Wang Zhi-Qiang
Published: March 2016

We present various results on multiple solutions for superlinear elliptic equations in a bounded domain or in the whole space. With a Nehari type condition assumed, we show that the standard Ambrosetti-Rabinowitz superlinear condition can be replaced by a more natural super-quadratic condition.
Existence of Multiple Positive Solutions for N-Laplacian in a Bounded Domain in ℝN
Authors: Prashanth S. , Sreenadh K.
Published: March 2016

Let Ω be a bounded domain in ℝN, N ≥ 2, λ > 0, q∊(0, N −1) and

In this article we show the existence of at least two positive solutions for the following quasilinear elliptic problem with an exponential type nonlinearity:

We use Monotonicity and Variational methods to obtain this multiplicity result.
Symmetrization and Mass Comparison for Degenerate Nonlinear Parabolic and Related Elliptic Equations
Authors: Vázquez Juan Luis
Published: March 2016

We consider the solutions to various nonlinear parabolic equations and their elliptic counterparts and prove comparison results based on two main tools, symmetrization and mass concentration comparison. The work focuses on equations like the porous medium equation, the filtration equation and the p-Laplacian equation. The results will be used in a companion work in combination with a detailed knowledge of special solutions to obtain sharp a priori bounds and decay estimates for wide classes of solutions of those equations.
Some Existence Results on Nontrivial Solutions of the Prescribed Mean Curvature Equation
Authors: Le Vy Khoi
Published: March 2016

The paper is concerned with an eigenvalue problem for the prescribed mean curvature equation. We formulate the problem as a variational inequality and show that under some growth conditions on the lower order term, the relaxed problem has at least two nontrivial solutions in a space of functions of bounded variation when the parameter is small.
Existence of Solutions for Some Classes of Singular Hamiltonian Systems
Authors: Alves Claudianor O. , Corrêa Francisco J. S. A. , Gonçalves Jose V. A.
Published: March 2016

This paper deals with existence of positive solution for the following singular Hamiltonian system

where Ω ⊂ ℝN , N ≥ 2, is a smooth bounded domain and H, T, S, K are continuous functions. A key role is played by the Galerkin method.
Rotation Numbers and Global Bifurcation in Systems of Ordinary Differential Equations
Authors: Ward James R.
Published: March 2016

Global bifurcation in boundary value problems for systems of ordinary differential equations is studied. Rotation numbers are associated with solutions and are shown to be invariant along bifurcating continua. This invariance is then used to analyze the global structure of the bifurcating continua, and to demonstrate the existence of infinitely many solutions to some boundary value problems.
Fixed Points for Dissipative-Repulsive Systems and Topological Dynamics of Mappings Defined on N-dimensional Cells
Authors: Pireddu Marina , Zanolin Fabio
Published: March 2016

We prove a fixed point theorem for continuous mappings which satisfy a compression-expansion condition on the boundary of a N-dimensional cell of ℝN. Our work is motivated by a previous theorem of Andres, Gaudenzi and Zanolin (1990) dealing with the existence of periodic solutions for dissipative-repulsive systems and is also related to some recent results by Zgliczyński and Gidea (2004) concerning covering relations for Markov partitions. Applications are given to the study of the iterates of a continuous map of ℝN. In this case we obtain some theorems on the existence of periodic points and chaotic-like dynamics.
Multiplicity of Entire Solutions For a Class of Almost Periodic Allen-Cahn Type Equations
Authors: Alessio Francesca , Montecchiari Piero
Published: March 2016

We consider a class of semilinear elliptic equations of the form

−Δu(x,y) + a(εx)Wʹ(u(x,y)) = 0, (x,y) ∊ ℝ2 (0.1)

where ε > 0, a : ℝ → ℝ is an almost periodic, positive function and W : ℝ → ℝ is modeled on the classical two well Ginzburg-Landau potential W(s) = (s2 - 1)2. We show via variational methods that if ε is sufficiently small and a is not constant then (0.1) admits infinitely many two dimensional entire solutions verifying the asymptotic conditions u(x, y) → ±1 as x → ±∞ uniformly with respect to y ∊ ℝ.
Bifurcation and Concentration of Radial Solutions of a Nonlinear Degenerate Elliptic Eigenvalue Problem
Authors: Evéquoz G. , Stuart C.A.
Published: March 2016

We consider the nonlinear boundary-value problem in ℝN (N ≥ 3),

where C ∊ C1 ([0, 1]) with C(r) > 0 for all r ∊ [0, 1]; C(0) = 0 and

and, for some T > 0; f ∊ C1([-T, T]) is an odd function that is strictly concave on [0, T] with f(0) = f(T) = 0 and f′(0) = 1. We prove that there is vertical bifurcation of radial solutions at every

in the sense that there exists a sequence {(λ, wn)} of solutions such that

These solutions concentrate at 0 in the sense that wn(0) = T for all n but wn converges uniformly to zero on all compact subsets that do not contain zero.
Asymptotic Behaviour of the Gradient of Large Solutions to Some Nonlinear Elliptic Equations
Authors: Alessio Porretta , Laurent Véron
Published: March 2016

If h is a nondecreasing real valued function and 0 ≤ q ≤ 2, we analyse the boundary behaviour of the gradient of any solution u of −Δu + h(u) + |∇u|q = f in a smooth N-dimensional domain Ω with the condition that u tends to infinity when x tends to ∂Ω. We give precise expressions of the blow-up which, in particular, point out the fact that the phenomenon occurs essentially in the normal direction to ∂Ω. Motivated by the blow-up argument in our proof, we also give a symmetry result for some related problems in the half space.
Regularity of Entropy Solutions of Quasilinear Elliptic Problems Related to Hardy-Sobolev Inequalities
Authors: Abdellaoui Boumediene , Colorado Eduardo , Sanchón Manel
Published: March 2016

This article is concerned with the regularity of the entropy solution of

where Ω is a smooth bounded domain Ω of ℝN such that 0 ∈ Ω, 1 < p < N. and γ < (N-p)/p. Assuming f ∈ Lq(Ω, |x|α(q-1)dx) for some q ≥ 1 and Nγp/(N-p) ≤ α ≤ (γ+1)p, we obtain estimates for the entropy solution u and its weak gradient in Lebesgue spaces with weights. Moreover, we introduce some explicit examples showing the optimality of our results and a relation between our problem and a Hardy-Sobolev type inequality.
Non-Degeneracy and Periodic Solutions of Semilinear Differential Equations with Deviation
Authors: Meng Gang , Yan Ping , Lin Xiaoyan , Zhang Meirong
Published: March 2016

In this paper, we will use some Sobolev constants to construct certain classes of nondegenerate potentials for linear differential equations with deviation. As an application of these potentials, we will give some existence results of periodic solutions of semilinear equations with deviation.
Isolated Singularities of Solutions of Quasilinear Anisotropic Elliptic Equations
Authors: Namlyeyeva Yuliya V. , Shishkov Andrey E. , Skrypnik Igor I.
Published: March 2016

We consider a wide class of degenerate quasilinear second order elliptic equations with model representative

, whose solutions have singularity at a point. There are established sharp point-wise conditions for removable isolated singularity of solutions of such equations. For solutions with non-removable point singularity (source type or fundamental solution), precise upper and lower estimates near the singularity point are obtained.
Comparison and Existence Results for Classes of Nonlinear Elliptic Equations with General Growth in the Gradient
Authors: Ferone Vincenzo , Messano Basilio
Published: March 2016

In this paper we study the Dirichlet problem for a class of nonlinear elliptic equations in the form A(u) = H(x, u, Du), where the principal term is a Leray-Lions operator defined on W0
1,p (Ω), and H(x, u, Du) grows with respect to Du at most like |Du|q, p-1 ≤ q ≤ p. Comparison results are obtained between the rearrangement of a solution u of Dirichlet problem quoted above and the rearrangement of the solution of a problem whose data are radially symmetric.
Topological Translations and Nonlinear Resonance
Authors: Dambrosio Walter , Ortega Rafael
Published: March 2016

In this paper we study resonance for nonlinear oscillators from a topological point of view. The Poincaré map associated to a linear second order equation at resonance is a translation of the plane. Our aim is to prove that in the case of nonlinear equations the Poincaré map is topologically conjugate to a translation. The proof is based on a characterization of translations in terms of Lyapunov functions.
Existence and Multiplicity of Positive Solutions For a Class of Elliptic Boundary Value Problems in the Half-Space
Authors: Hsu Tsing-San
Published: March 2016

In this paper, we prove that if b(x) satisfies some suitable conditions, then -Δu + u = b(x)up in ℝN
+ with the boundary condition u(xʹ, 0) = λg(xʹ) has at least two positive solutions if 0 < λ < λ*, a minimal positive solution if λ = λ* and no positive solution if λ > λ*.
Back to the Keller-Osserman Condition for Boundary Blow-up Solutions
Authors: Dumont Serge , Dupaigne Louis , Goubet Olivier , Rădulescu Vicentiu
Published: March 2016

This article is concerned with the existence, uniqueness and numerical approximation of boundary blow up solutions for elliptic PDE’s Δu = f(u), where f satisfies the so-called Keller-Osserman condition. We characterize existence of such solutions for non-monotone f. As an example, we construct an infinite family of boundary blow up solutions for the equation Δu = u2(1 + cos u) on a ball. We prove uniqueness (on balls) when f is increasing and convex in a neighborhood of infinity and we discuss and perform some numerical computations to approximate such boundary blow-up solutions.
On Algebro-Geometric Solutions of the Camassa-Holm Hierarchy
Authors: Zampogni Luca
Published: March 2016

We find global solutions of algebro geometric type for all the equations of a new commuting hierarchy containing the Camassa-Holm equation. This hierarchy is built in analogy to the classical K-dV and AKNS hierarchies. We use a zero curvature method to give recursion formulas. The time evolution of the solutions is completely determined, and the motion on a nonlinear subvariety Υ of a generalized Jacobian variety is obtained by solving an inverse problem for the Sturm-Liouville equation L(φ) = −φ″ + φ = λyφ. This is the natural setting for the expression of the solutions which depend linearly with respect to t and x, with coordinates on a curvilinear parallelogram contained in such a subvariety φ. φ is obtained as the restriction of the generalized Abel map I0 to the space Symmg(R) of unordered g-tuples of points on R, and the nonlinear parallelogram is the image through the restricted generalized Abel map of the moving poles Pi(x, t) (i = 1, . . . , g) of the Weyl m-functions m±(x, t, λ) of the dynamical Sturm-Liouville family of equations Lx(φ) = −φ″ + φ = λτx(y)φ, where τ is the translation flow. It turns out that the choice of a particular stationary initial condition ug(x, t0) completely determines the solution u(x, t) of all the equations in the hierarchy, as functions of the poles Pi(x, t) of the Weyl m-functions corresponding to the family Lx of Sturm-Liouville operators, with density function y(x, t) = uxx(x, t)/2 − 2u(x, t). For every t∊ℝ, the maps x ↦ y(x, t) lie in an isospectral class of the associated family of Sturm-Liouville equations Lx(φ), and are completely determined by assigning spectral parameters and initial conditions for the poles Pi(x, t).
Existence and Stability of Standing Waves For Schrödinger-Poisson-Slater Equation
Authors: Kikuchi Hiroaki
Published: March 2016

We study the existence and stability of standing wave for the Schrödinger-Poisson-Slater equation in three dimensional space. Let p be the exponent of the nonlinear term. Then we first show that standing wave exists for 1 < p < 5. Next, we show that when 1 < p < 7/3 and p ≠ 2, standing wave is stable for some ω > 0. We also show that when 7/3 < p < 5, standing wave is unstable for some ω > 0. Furthermore, we investigate the case of p = 2. We prove these results by using variational methods.
On Existence of L∞-Ground States for Singular Elliptic Equations in the Presence of a Strongly Nonlinear Term
Authors: Goncalves J.V. , Melo A.L. , Santos C.A.
Published: March 2016

We establish new results concerning existence and the behavior at infinity of solutions for the singular nonlinear elliptic equation −Δu = ρa(x)u−α + λb(x)uγ in ℝN, where a, b are nonnegative Hölder continuous functions with a + b > 0, α,γ > 0 are constants and ρ, γ are positive parameters. We emphasize that powers uγ with γ ≤ 1 were treated earlier by a number of authors. Our results address mainly terms uγ, where γ is any number bigger than one which we name strongly nonlinear. Penalty arguments, variational principles and an approximation procedure will be exploited.
Global Existence for Quadratic Systems of Reaction-Diffusion
Authors: Desvillettes Laurent , Fellner Klemens , Pierre Michel , Vovelle Julien
Published: March 2016

We prove global existence in time of weak solutions to a class of quadratic reaction-diffusion systems for which a Lyapounov structure of L log L-entropy type holds. The approach relies on an a priori dimension-independent L2-estimate, valid for a wider class of systems including also some classical Lotka-Volterra systems, and which provides an L1-bound on the nonlinearities, at least for not too degenerate diffusions. In the more degenerate case, some global existence may be stated with the use of a weaker notion of renormalized solution with defect measure, arising in the theory of kinetic equations.
A Multiplicity Result Including Sign-Changing Solutions For a Nonlinear Elliptic Problem in RN
Authors: Hirano Norimichi , Shioji Naoki
Published: March 2016

We show that the problem

has multiple solutions including sign-changing ones, where μ > 0, N ≥ 3, 1 < p < (N + 2)/(N − 2) and Q : ℝN → ℝ is some continuous function. We obtain signchanging solutions which are not only local minimum types in N* but also mountain pass types in N*, where N* = {u = u+ +u− ∈ H1(ℝN) : u+ ≠ 0, u− ≠ 0, ∫ℝN (|∇u±|2 + μ|u±|2) dx = ∫ℝN Q(x)|u±|p+1 dx}.
On a Minimization Problem Involving the Critical Sobolev Exponent
Authors: Prinari Francesca , Visciglia Nicola
Published: March 2016

Following [3] we study the following minimization problem:

in any dimension n ≥ 4 and under suitable assumptions on a(x). Mainly we assume that a(x) belongs to the Lorentz space Ln/2 ,d(Ω) and the set

N ≡ {x ∈ Ω|a(x) < 0}

has positive Lebesgue measure. Notice that this last condition is satisfied when the set N has a nontrivial interior part (in fact this is the typical assumption imposed in the literature on the set N).
Soliton Solutions to a Class of Quasilinear Elliptic Equations on ℝ
Authors: Alves M.J. , Carrião P.C. , Miyagaki O.H.
Published: March 2016

This paper is concerned with the existence of positive solutions for a class of quasilinear elliptic equations on ℝ. The results are proved by combining the concentration-compactness principle due to Lions with a minimization approach.
Multiplicity of Solutions For p-Laplacian Equations Which Need Not be Coercive
Authors: Filippakis Michael E. , Papageorgiou Nikolaos S.
Published: March 2016

In this paper, we consider nonlinear Dirichlet problem driven by the p-Laplacian differential operator. Using variational methods based on the critical point theory and truncation techniques, we prove the existence of at least three nontrivial smooth solutions. The hypotheses on the nonlinearity incorporate in our framework of analysis both coercive and noncoercive problems. For the semilinear problem (p = 2), using Morse theory, we show the existence of four nontrivial smooth solutions.
Positive and Nodal Solutions For a Nonlinear Schrödinger Equation with Indefinite Potential
Authors: Furtado Marcelo F. , Maiay Liliane A. , Medeiros Everaldo S.
Published: March 2016

We deal with the nonlinear Schrödinger equation

-Δu + V(x)u = f(u) in ℝN,

where V is a (possible) sign changing potential satisfying mild assumptions and the nonlinearity f ∈ C1(ℝ, ℝ) is a subcritical and superlinear function. By combining variational techniques and the concentration-compactness principle we obtain a positive ground state solution and also a nodal solution. The proofs rely in localizing the infimum of the associated functional constrained to Nehari type sets.
Asymptotic Behavior of The Unique Solution to a Singular Elliptic Problem With Nonlinear Convection Term And Singular Weight
Authors: Zhang Zhijun
Published: March 2016

By Karamata regular variation theory, we first derived the exact asymptotic behavior of the local solution to the problem -φʹʹ(s) = g(φ(s)), φ(s) > 0, s ∈ (0, a) and φ(0) = 0. Then, by a perturbation method and constructing comparison functions, we derived the exact asymptotic behavior of the unique classical solution near the boundary to a singular Dirichlet problem -Δu = b(x)g(u) + λ|▽u|q, u > 0, x ∈ Ω, u|aΩ = 0, where Ω is a bounded domain with smooth boundary, λ ∈ ℝ, q ∈ [0, 2]; g ∈ C1((0, ∞), (0, ∞)), is decreasing in (0, ∞) with lims→0 + g(s) = +∞; the weight b is positive in Ω and singular on the boundary.
A Note on Berestycki-Cazenave's Classical Instability Result for Nonlinear Schrödinger Equations
Authors: Le Coz Stefan
Published: March 2016

In this note we give an alternative, shorter proof of the classical result of Berestycki and Cazenave on the instability by blow-up for the standing waves of some nonlinear Schrödinger equations.
Morse Indices of Closed Geodesics on Katok's 2-Spheres
Authors: Long Yiming , Wang Wei
Published: March 2016

In this paper, we compute precisely Morse indices of iterates of the two closed geodesics on the 2-sphere with Katok’s metric. This information is used to give a direct proof for the Poincaré series of the free loop space on S2.
The Multidimensional Bipolar Quantum Drift-diffusion Model
Authors: Chen Xiuqing , Guo Yingchun
Published: March 2016

We investigate the bipolar quantum drift-diffusion model, a fourth order parabolic system, in two or there space dimensions. First, we establish the global weak solution with large initial value and periodic boundary conditions. Then we show the semiclassical limit by using new methods based on delicate interpolation estimates and compactness argument. Furthermore, when the doping profile is a constant, we find that the weak solution approaches its mean value exponentially as time increases to infinity.
Vortices with Prescribed L2Norm in the Nonlinear Wave Equation
Authors: Badiale Marino , Rolando Sergio
Published: March 2016

We prove the existence of rotating solitary waves (vortices) with enough largely prescribed L2norm for the nonlinear Klein-Gordon equation with nonnegative potential, which makes the equation suitable for physical models and guarantees the well-posedness of the corresponding Cauchy problem. This is done by finding nonnegative cylindrical solutions to the standing equation

where x = (y, z) ∈ ℝk× RN–k, N > k ≥ 0 and μ > 0 is the fundamental eigenvalue, namely, the Lagrange multiplier of the minimization problem

(with infimum over a suitable subspace of H1(ℝN)).
Existence and Multiplicity of Solutions for Neumann p-Laplacian-Type Equations
Authors: Gasiński Leszek , Papageorgiou Nikolaos S.
Published: March 2016

We consider nonlinear Neumann problems driven by p-Laplacian-type operators which are not homogeneous in general. We prove an existence and a multiplicity result for such problems. In the existence theorem, we assume that the right hand side nonlinearity is p-superlinear but need not satisfy the Ambrosetti-Rabinowitz condition. In the multiplicity result, when specialized to the case of the p-Laplacian, we allow strong resonance at infinity and resonance at 0.
The Porous Medium Equation With Blowing Up Boundary Data
Authors: T. Osvaldo Venegas
Published: March 2016

We study the blow-up set of a solution u(x, t) to the porous medium equation, ut = Δ(um), in Ω × (0, T) for two different boundary conditions, u(x, t) = f(x)/(T - t)

or

(x, t) = g(x)/(T - t)

on ӘΩ × (0, T). Here m > 1 and Ω is a bounded smooth domain in ℝN. We establish point-wise and integral conditions on f and g respectively in order to obtain regional or global blow-up.
On Uniqueness of Large Solutions of Nonlinear Parabolic Equations in Nonsmooth Domains
Authors: Al Sayed Waad , Véron Laurent
Published: March 2016

We study the existence and uniqueness of the positive solutions of the problem (P): Әtu - Δu + uq = 0 (q > 1) in Ω × (0, ∞), u = ∞ on ӘΩ × (0, ∞) and u(., 0) ∈ L1(Ω), when Ω is a bounded domain in ℝN. We construct a maximal solution, prove that this maximal solution is a large solution whenever q < N/(N - 2) and it is unique if ӘΩ = ӘΩ̅c. If ӘΩ has the local graph property, we prove that there exists at most one solution to problem (P).
Asymptotic Analysis of a Radionuclide Transport Model With Unbounded Viscosity
Authors: Choquet Catherine
Published: March 2016

We consider a model describing compressible nuclear waste disposal contamination in porous media. The transport of brine and radionuclides is described by a nonlinear coupled degenerate parabolic system. The viscosity of the fluid is unbounded and concentrations and temperature dependent. We study the asymptotic behavior of the model for little Peclet numbers.
Periodic Perturbations With Delay of Autonomous Differential Equations on Manifolds
Authors: Furi Massimo , Spadini Marco
Published: March 2016

We apply topological methods to the study of the set of harmonic solutions of periodically perturbed autonomous ordinary differential equations on differentiable manifolds, allowing the perturbing term to contain a fixed delay. In the crucial step, in order to cope with the delay, we define a suitable (infinite dimensional) notion of Poincaré T-translation operator and prove a formula that, in the unperturbed case, allows the computation of its fixed point index.
Multiplicity of Positive Solutions for Critical Singular Elliptic Systems With Concave-Convex Nonlinearities
Authors: Hsu Tsing-San
Published: March 2016

In this paper, we consider a singular elliptic system with both concave-convex nonlinearities and critical growth terms in bounded domains. The existence and multiplicity results of positive solutions are obtained by variational methods.
Maximum Principles for Inhomogeneous Elliptic Inequalities on Complete Riemannian Manifolds
Authors: Mugnai Dimitri , Pucci Patrizia
Published: March 2016

We prove some maximum principle results for weak solutions of elliptic inequalities, possibly inhomogeneous, on complete Riemannian manifolds.
Periodic and Bloch Solutions to a Magnetic Nonlinear Schrödinger Equation
Authors: Clapp Mónica , Iturriaga Renato , Szulkin Andrzej
Published: March 2016

We study the equation

(℘A) (-i▽ + A)2u + V u = |u|p-2u,

where A ∈C1,α(ℝN,ℝN) and V ∈C0,α(ℝN) are 2π-periodic in each variable, V > 0; and p ∈(2; 2*) with 2* := ∞ if N = 2 and

We address two questions: first, the gauge-dependence problem for 2π-periodic solutions u : ℝN → C and second, the multiplicity of Bloch solutions. Unlike the nonperiodic case where problem (℘A) is basically independent of A (it is gauge invariant), in the periodic case this is far from being true. Under some assumptions on A we show that, if there exists a one-to-one correspondence between the 2π-periodic solutions of (℘A) and those of (℘A + z) preserving their absolute value, then z lies in a subset of measure zero of ℝN. We use this fact to show the existence of an uncountable set of Bloch solutions with real quasimomentum.
Global Continuation for Quasilinear Elliptic Systems on ℝN and the Equations of Elastostatics
Authors: Gebran Hicham G. , Stuart Charles A.
Published: March 2016

We consider quasilinear systems of second order elliptic equations on ℝN. Using a continuation theorem based on the topological degree for C1 - Fredholm maps, we derive global properties of a maximal connected set of solutions which decay exponentially to zero at in¯nity. These results are used to treat a problem concerning the equilibrium of an elastic body occupying the whole space and subjected to a one parameter family of localized external forces.
A Note on a Class of Reversible Hamiltonian Systems
Authors: Rabinowitz Paul H.
Published: March 2016

Earlier papers have studied the existence of solutions of reversible Hamiltonian systems that are heteroclinic between a pair of minimizing periodic solutions. This note extends some of that work by allowing more degenerate behavior for the class of periodic minimizers.
On Some Quasilinear Critical Problems
Authors: Valeriola Sébastien de , Willem Michel
Published: March 2016

In this paper we prove the almost everywhere convergence of the gradient of Palais-Smale sequences, allowing us to apply the Brezis-Lieb lemma. This leads us to show that infima are attained, and thus to prove the existence of optimal solutions for some critical problems. Our method does not use the concentration- compactness principle.
Ground States and Bound States of a Nonlinear Schrödinger System
Authors: Liu Zhaoli , Wang Zhi-Qiang
Published: March 2016

This paper concerns existence and multiplicity of ground states and bound states of the time-independent Schrödinger system

where n = 2, 3, N ≥ 2, λj > 0 for j = 1, ... , N, βjj > 0 for j = 1, ... , N,

and βij = βji. In the attractive case we give sufficient condition for existence of co-existing ground states with large couplings, and in the repulsive case we prove existence of infinitely many co-existing bound state solutions with arbitrary couplings.
Addendum to the Paper “Unique Strong Solutions and V -Attractors of a Three Dimensional System of Globally Modified Navier-Stokes Equations″, Advanced Nonlinear Studies 6 (2006), 411-436
Authors: Caraballo Tomás , Real José , Kloeden Peter E.
Published: March 2016

In this paper we improve Theorem 7 in [1] which deals with the existence and uniqueness of solutions of the three dimensional globally modified Navier-Stokes equations.
Boundary Behavior of Solutions to Singular Boundary Value Problems for Nonlinear Elliptic Equations
Authors: Zhang Zhijun , Li Xiaohong , Zhao Yuanzhang
Published: March 2016

Let Ω be a bounded domain with smooth boundary in ℝN and h ∈ C((0,∞), (0,∞)) with lims→0+ h(s) = Υ ∈ (0,∞). By the perturbation method, which is due to García Melián, and nonlinear transformations and comparison principles, we derive the exact boundary behavior of solutions to a singular Dirichlet problem

. Then, applying the result, combining two kinds of nonlinear transformations, we derive the exact boundary behavior of solutions to a boundary blow-up elliptic problem and a singular Dirichlet problem, where the weight b is positive in Ω and may be (rapidly) vanishing or blow up on the boundary.
Existence and Concentration of Bound States for a p-Laplacian Equation in ℝN
Authors: Gloss Elisandra
Published: March 2016

We study existence and asymptotic behavior of positive solutions for quasilinear elliptic equations of the form -εpΔpv + V(x)|v|p-2v = f(v) in ℝN, 1 < p < N, as ε → 0. The potential V : ℝN → ℝ has a positive infimum, inf∂ΩV > infΩ V for some bounded domain Ω in ℝN and f is a subcritical nonlinearity without some growth conditions such as Ambrosetti-Rabinowitz.
Global Existence and Blowup for the Cubic Nonlinear Klein-Gordon Equations in Three Space Dimensions
Authors: Zhang Jian , Gan Zaihui , Guo Boling
Published: March 2016

In this paper, we apply a cross-constrained variational method to study the classic nonlinear Klein-Gordon equation with cubic nonlinearity in three space dimensions. By constructing a type of cross-constrained variational problem and establishing the so-called cross invariant manifolds, we obtain a sharp threshold for blowup and global existence of the solution to the equation under study which is different from that in [10] . On the other hand, we give an answer to the question that how small the initial data have to be for the global solutions to exist.
Solitons for the Nonlinear Klein-Gordon Equation
Authors: Bellazzini J. , Benci V. , Bonanno C. , Micheletti A.M.
Published: March 2016

In this paper we study existence and orbital stability for solitary waves of the nonlinear Klein-Gordon equation. The energy of these solutions travels as a localized packet, hence they are a particular type of solitons. In particular we are interested in sufficient conditions on the potential for the existence of solitons. Our proof is based on the study of the ratio energy/charge of a function, which turns out to be a useful approach for many field equations.
Bifurcation of Limit Cycles from a Polynomial Degenerate Center
Authors: Buică Adriana , Giné Jaume , Llibre Jaume
Published: March 2016

Using Melnikov functions at any order, we provide upper bounds for the maximum number of limit cycles bifurcating from the period annulus of the degenerate center ẋ = −y((x2 + y2)/2)m and ẏ = x((x2 + y2)/2)m with m ≥ 1, when we perturb it inside the whole class of polynomial vector fields of degree n. The positive integers m and n are arbitrary. As far as we know there is only one paper that provide a similar result working with Melnikov functions at any order and perturbing the linear center ẋ = −y, ẏ = x.
Symmetry Results for Nonvariational Quasi-Linear Elliptic Systems
Authors: Montoro Luigi , Sciunzi Berardino , Squassina Marco
Published: March 2016

By virtue of a weak comparison principle in small domains we prove axial symmetry in convex and symmetric smooth bounded domains as well as radial symmetry in balls for regular solutions of a class of quasi-linear elliptic systems in non-variational form. Moreover, in the two dimensional case, we study the system when set in a half-space.
A Minimum Problem with Free Boundary for the p(x)−Laplace Operator
Authors: Lyaghfouri A.
Published: March 2016

In this paper we consider the problem of minimizing the functional

We prove Lipschitz continuity for each minimizer u and establish the nondegeneracy at the free boundary (∂[u > 0]) ∩ Ω and the locally uniform positive density of the sets [u > 0] and [u = 0]. In particular we obtain that the Lebesgue measure of the free boundary is zero.
Subharmonic Solutions of Planar Hamiltonian Systems: a Rotation Number Approach
Authors: Boscaggin Alberto
Published: March 2016

We prove the existence of infinitely many subharmonic solutions, with prescribed nodal properties, for a planar Hamiltonian system Jz′ = ΔzH(t, z), with H periodic in the first variable. The goal is achieved by performing estimates of the rotation numbers with respect to deformed polar coordinates and applying Ding’s version of the Poincaré-Birkhoff fixed point theorem.
A Legendre Transform on an Exotic S3
Authors: Martino Vittorio
Published: March 2016

We consider an exotic contact form α on S3 and we establish explicitly the existence of a non singular vector field v in ker(α) such that the non-singular one-differential form β(・) := dα(v, ・) is a contact form on S3 with the same orientation as α. In particular this means that a Legendre transform can be completed.
Regularizing Effect of a Gradient Term in a Problem Involving the p-Laplacian Operator
Authors: Perrotta Adamaria , Primo Ana
Published: March 2016

In this article we consider the following problem

with max{p − 1, 1} ≤ q ≤ p < N, Ω a bounded domain and f, g positive measurable functions in Ω. We give assumptions on g with respect to q and p for which for all λ > 0 and all f ∈ L1(Ω), f ≥ 0, problem (P) has a positive solution. In particular, we focus our attention on the case when 0 ∈ Ω and

to prove that the assumptions on g are optimal. Furthermore, regularity results are given.
Multiplicity and Concentration of Positive Solutions for a Class of Quasilinear Problems
Authors: Alves Claudianor O. , Figueiredo Giovany M.
Published: March 2016

In this work we study multiplicity and concentration of nontrivial solutions for the following class of quasilinear problems:

-div(a(|▽u|p)|▽u|p-2▽u) + V(εx)b(|u|p)|u|p-2u=f(u),

where ε is a positive parameter, a, b, f : ℝ → ℝ are C1 functions and V : ℝN →ℝ is a continuous function.
Homogenization of Maximal Monotone Vector Fields via Selfdual Variational Calculus
Authors: Ghoussoub Nassif , Moameni Abbas , Sáiz Ramón Zárate
Published: March 2016

We use the theory of selfdual Lagrangians to give a variational approach to the homogenization of equations in divergence form, that are driven by a periodic family of maximal monotone vector fields. The approach has the advantage of using Γ-convergence methods for corresponding functionals just as in the classical case of convex potentials, as opposed to the graph convergence methods used in the absence of potentials. A new variational formulation for the homogenized equation is also given.
Nonlinear Resonance: a Comparison Between Landesman-Lazer and Ahmad-Lazer-Paul Conditions
Authors: Fonda Alessandro , Garrione Maurizio
Published: March 2016

We show that the Ahmad-Lazer-Paul condition for resonant problems is more general than the Landesman-Lazer one, discussing some relations with other existence conditions, as well. As a consequence, such a relation holds, for example, when considering resonant boundary value problems associated with linear elliptic operators, the p-Laplacian and, in the scalar case, with an asymmetric oscillator.
A Uniqueness Result for Δu - λu + V(⃒x⃒)uP = 0 on ℝ2
Authors: Genoud François
Published: March 2016

Uniqueness of positive radial solutions decaying at infinity is proved for a class of semilinear elliptic equations on ℝ2. Complementary results for the same kind of equations were obtained in the early 90’s, on ℝN with N ≥ 3, and in finite balls of ℝN with N ≥ 2. The new result presented here plays a crucial role in the global bifurcation problem, previously studied by the author.
Quasilinear Elliptic Problems with General Growth and Nonlinear Term Having Singular Behavior
Authors: El-Hadi Miri Sofiane
Published: March 2016

In this paper we deal with the following quasilinear elliptic problems

where Ω ⊂ ℝNis a bounded regular domain, d(x) = dist(x, ∂Ω), p > 1, λ > 0 and 0 < ν ≤ p. Moreover, g, q and f are nonnegative functions verifying suitable hypotheses. The main goal of this work is to analyze the interaction between the functions d, g and the gradient term to get existence and nonexistence results. The singular behavior of g near 0 will provide a new phenomenon that does not appear in the case where the absorption term g(u) q(d(x)) is dropped or substituted by a regular one.
Radial Solutions of a Supercritical Elliptic Equation with Hardy Potential
Authors: Guo Zuji , Liu Zhaoli
Published: March 2016

Various properties of radial solutions of the supercritical elliptic equation with Hardy Potential

are studied, where Ω = int{x ∈ ℝN | α ≤ |x| < b}, which is a ball if 0 = α < b < +∞, an annulus if 0 < α < b < +∞, an exterior domain if 0 < α < b = +∞, and the whole space ℝN if α = 0, b = +∞. We assume p is supercritical, that is, p > 2∗ with 2∗ =

being the critical Sobolev exponent, and N ≥ 3.
Strong Maximum Principles for Anisotropic Elliptic and Parabolic Equations
Authors: Vétois Jérôme
Published: March 2016

We investigate vanishing properties of nonnegative solutions of anisotropic elliptic and parabolic equations. We describe the optimal vanishing sets, and we establish strong maximum principles.
Count and Symmetry of Global and Local Minimizers of the Cahn-Hilliard Energy Over Cylindrical Domains
Authors: Nascimento Arnaldo Simal do , Biesdorf João , Crema Janete
Published: March 2016

We address the problem of minimization of the Cahn-Hilliard energy functional under a mass constraint over two and three-dimensional cylindrical domains. Although existence is presented for a general case the focus is mainly on rectangles, parallelepipeds and circular cylinders. According to the symmetry of the domain the exact numbers of global and local minimizers are given as well as their geometric profile and interface locations; all are onedimensional increasing/decreasing and odd functions for domains with lateral symmetry in all axes and also for circular cylinders. The selection of global minimizers by the energy functional is made via the smallest interface area criterion.

The approach utilizes Γ−convergence techniques to prove existence of an one-parameter family of local minimizers of the energy functional for any cylindrical domain. The exact numbers of global and local minimizers as well as their geometric profiles are accomplished via suitable applications of the unique continuation principle while exploring the domain geometry in each case and also the preservation of global minimizers through the process of Γ−convergence.
Min-Max Solutions to Some Scalar Field Equations
Authors: Devillanova Giuseppe , Solimini Sergio
Published: March 2016

We show the variational structure of a multiplicity result of positive solutions u ∈ H1(ℝN) to the equation −Δu + a(x)u = up, where N ≥ 2, p > 1 with p < 2∗ − 1 =

and the potential a(x) is a positive function enjoying a planar symmetry. We require suitable decay assumptions which are widely implied by those in [6], in which Wei and Yan have obtained an analogous multiplicity result by using different techniques.
Sublinear Elliptic Equations With Singular Potentials
Authors: Boccardo Lucio , Orsina Luigi
Published: March 2016

In this paper we prove existence and summability results for solutions of equations whose model is

with a > 0 and f in some Marcinkiewicz space, thus extending the results of [4]. We the use these results to prove similar theorems for equations whose model is

with 0 < θ < 1 and p in some Marcinkiewicz space, thus extending the results of [3].
Asymptotic Properties of Ground States of Quasilinear Schrödinger Equations With H1-Subcritical Exponent
Authors: Adachi Shinji , Watanabe Tatsuya
Published: March 2016

We are concerned with the asymptotic behavior of positive radial solutions for a class of quasilinear elliptic equation arising from plasma physics. By the variational argument and dual approach, we show the asymptotic uniqueness and non-degeneracy of the ground state.
Entire Solutions to Nonlinear Scalar Field Equations with Indefinite Linear Part
Authors: Evéquoz Gilles , Weth Tobias
Published: March 2016

We consider the stationary semilinear Schrödinger equation

−Δu + a(x)u = f (x, u), u ∈ H1(ℝN),

where a and f are continuous functions converging to some limits a∞ > 0 and f∞ = f∞(u) as |x| → ∞. In the indefinite setting where the Schrödinger operator −Δ + a has negative eigenvalues, we combine a reduction method with a topological argument to prove the existence of a solution of our problem under weak one-sided asymptotic estimates. The minimal energy level need not be attained in this case. In a second part of the paper, we prove the existence of ground-state solutions under more restrictive assumptions on a and f . We stress that for some of our results we also allow zero to lie in the spectrum of −Δ+a.
One-Signed Harmonic Solutions and Sign-Changing Subharmonic Solutions to Scalar Second Order Differential Equations
Authors: Boscaggin Alberto
Published: March 2016

Using a recent modified version of the Poincaré-Birkhoff fixed point theorem [19], we study the existence of one-signed T-periodic solutions and sign-changing subharmonic solutions to the second order scalar ODE

u′′ + f (t, u) = 0,

being f : ℝ × ℝ → ℝ a continuous function T-periodic in the first variable and such that f (t, 0) ≡ 0. Partial extensions of the results to a general planar Hamiltonian systems are given, as well.
An Example of Chaos for a Cubic Nonlinear Schrödinger Equation with Periodic Inhomogeneous Nonlinearity
Authors: Zanini Chiara , Zanolin Fabio
Published: March 2016

Using a topological approach we prove the existence of infinitely many periodic solutions, as well as the presence of symbolic dynamics for second order nonlinear differential equations of the form

−ü − μu + g(t)h(u) = 0

where μ > 0 is a given constant and g : ℝ → ℝ is a periodic positive weight function. Our main application concerns the study of the case in which h(u) is a cubic nonlinearity. Such a choice is motivated by previous investigations dealing with the nonlinear Schrödinger equation
On the Existence and Breaking Symmetry of the Ground State Solution of Hardy Sobolev Type Equations withWeighted p-LAPLACIAN
Authors: Bhakta Mousomi
Published: March 2016

We study the existence of ground state solution, cylindrically symmetric solution, breaking cylindrical symmetry of ground state solution of the following Hardy-Sobolev type equa- tion with weighted p-Laplacian:

where
Positive Solutions of the Dirichlet Problem for the One-dimensional Minkowski-Curvature Equation
Authors: Coelho Isabel , Corsato Chiara , Obersnel Franco , Omari Pierpaolo
Published: March 2016

We discuss existence and multiplicity of positive solutions of the Dirichlet problem for the quasilinear ordinary differential equation

.

Depending on the behaviour of f = f (t, s) near s = 0, we prove the existence of either one, or two, or three, or infinitely many positive solutions. In general, the positivity of f is not required. All results are obtained by reduction to an equivalent non-singular problem to which variational or topological methods apply in a classical fashion.