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Communications on Pure & Applied Analysis

November 2017 , Volume 16 , Issue 6

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On a class of rotationally symmetric $p$-harmonic maps
L. F. Cheung, C. K. Law and M. C. Leung
2017, 16(6): 1941-1955 doi: 10.3934/cpaa.2017095 +[Abstract](649) +[HTML](1) +[PDF](137.5KB)
Abstract:

We give a classification of rotationally symmetric \begin{document} $p$ \end{document}-harmonic maps between some model spaces such as \begin{document} $\mathbb{R}^n$ \end{document} and \begin{document} $\mathbb{H}^n$ \end{document} by their asymptotic behaviors. Among others, we show that, when \begin{document} $p>2$ \end{document} and \begin{document} $n≥q 2$ \end{document}, all rotationally symmetric \begin{document} $p$ \end{document}-harmonic maps from \begin{document} $\mathbb{R}^n$ \end{document} to \begin{document} $\mathbb{H}^n$ \end{document} have to blow up at a finite point, while all rotationally symmetric \begin{document} $p$ \end{document}-harmonic maps from \begin{document} $\mathbb{H}^n$ \end{document} to \begin{document} $\mathbb{H}^n$ \end{document} observe the trichotomy property, i.e. the map \begin{document} $y$ \end{document} is the identity map, is bounded or blows up according as its initial value \begin{document} $y'(0)$ \end{document} is equal to, less than or greater than one. Our sharp estimates imply and improve a number of existence and non-existence results of certain \begin{document} $p$ \end{document}-harmonic maps on noncompact manifolds.

Multiple complex-valued solutions for the nonlinear Schrödinger equations involving magnetic potentials
Gan Lu and Weiming Liu
2017, 16(6): 1957-1975 doi: 10.3934/cpaa.2017096 +[Abstract](736) +[HTML](2) +[PDF](462.8KB)
Abstract:

This paper is concerned with the following nonlinear Schrödinger equations with magnetic potentials

\begin{document}$\begin{equation}\label{0} \Bigl(\frac{\nabla}{i}-α A(|x|)\Bigl)^{2}u+(1+α V(|x|))u=|u|^{p-2}u,\,\,u∈ H^{1}(\mathbb{R}^{N},\mathbb{C}),\ \ \ \ \ \ \ \ \ \ \left( 0.1 \right) \end{equation}$\end{document}

where \begin{document} $2<p<\frac{2N}{N-2}$ \end{document} if \begin{document} $N≥q 3$ \end{document} and \begin{document} $2<p<+∞$ \end{document} if \begin{document} $N=2$ \end{document}. \begin{document} $α$ \end{document} can be regarded as a parameter. \begin{document} $A(|x|)=(A_{1}(|x|),A_{2}(|x|),···,A_{N}(|x|))$ \end{document} is a magnetic field satisfying that \begin{document} $A_{j}(|x|)>0(j=1,...,N)$ \end{document} is a real \begin{document} $C^{1}$ \end{document} bounded function on \begin{document} $\mathbb{R}^{N}$ \end{document} and \begin{document} $V(|x|)>0$ \end{document} is a real continuous electric potential. Under some decaying conditions of both electric and magnetic potentials which are given in section 1, we prove that the equation has multiple complex-valued solutions by applying the finite reduction method.

Almost reducibility of linear difference systems from a spectral point of view
Álvaro Castañeda and Gonzalo Robledo
2017, 16(6): 1977-1988 doi: 10.3934/cpaa.2017097 +[Abstract](755) +[HTML](4) +[PDF](125.8KB)
Abstract:

We prove that, under some conditions, a linear nonautonomous difference system is Bylov's almost reducible to a diagonal one whose terms are contained in the Sacker and Sell spectrum of the original system.

In the above context, we provide an example of the concept of diagonally significant system, recently introduced by Pötzsche. This example plays an essential role in the demonstration of our results.

The Green function for the Stokes system with measurable coefficients
Jongkeun Choi and Ki-Ahm Lee
2017, 16(6): 1989-2022 doi: 10.3934/cpaa.2017098 +[Abstract](803) +[HTML](4) +[PDF](596.5KB)
Abstract:

We study the Green function for the stationary Stokes system with bounded measurable coefficients in a bounded Lipschitz domain \begin{document} $Ω\subset \mathbb{R}^n$ \end{document}, \begin{document} $n≥ 3$ \end{document}. We construct the Green function in \begin{document} $Ω$ \end{document} under the condition \begin{document} $(\bf{A1})$ \end{document} that weak solutions of the system enjoy interior Hölder continuity. We also prove that \begin{document} $(\bf{A1})$ \end{document} holds, for example, when the coefficients are \begin{document} $\mathrm{VMO}$ \end{document}. Moreover, we obtain the global pointwise estimate for the Green function under the additional assumption \begin{document} $(\bf{A2})$ \end{document} that weak solutions of Dirichlet problems are locally bounded up to the boundary of the domain. By proving a priori \begin{document} $L^q$ \end{document}-estimates for Stokes systems with \begin{document} $\mathrm{BMO}$ \end{document} coefficients on a Reifenberg domain, we verify that \begin{document} $(\bf{A2})$ \end{document} is satisfied when the coefficients are \begin{document} $\mathrm{VMO}$ \end{document} and \begin{document} $Ω$ \end{document} is a bounded \begin{document} $C^1$ \end{document} domain.

Two-and multi-phase quadrature surfaces
Avetik Arakelyan, Henrik Shahgholian and Jyotshana V. Prajapat
2017, 16(6): 2023-2045 doi: 10.3934/cpaa.2017099 +[Abstract](680) +[HTML](17) +[PDF](504.9KB)
Abstract:

In this paper we shall initiate the study of the two-and multi-phase quadrature surfaces (QS), which amounts to a two/multi-phase free boundary problems of Bernoulli type. The problem is studied mostly from a potential theoretic point of view that (for two-phase case) relates to integral representation

where \begin{document} $dσ_x$ \end{document} is the surface measure, \begin{document} $μ= μ^+ - μ^-$ \end{document} is given measure with support in (a priori unknown domain) \begin{document} $Ω=Ω^+\cupΩ^-$ \end{document}, \begin{document} $g$ \end{document} is a given smooth positive function, and the integral holds for all functions \begin{document} $h$ \end{document}, which are harmonic on \begin{document} $\overline Ω$ \end{document}.

Our approach is based on minimization of the corresponding two-and multi-phase functional and the use of its one-phase version as a barrier. We prove several results concerning existence, qualitative behavior, and regularity theory for solutions. A central result in our study states that three or more junction points do not appear.

Sharp Strichartz estimates in spherical coordinates
Robert Schippa
2017, 16(6): 2047-2051 doi: 10.3934/cpaa.2017100 +[Abstract](688) +[HTML](5) +[PDF](314.2KB)
Abstract:

We prove Strichartz estimates found after adding regularity in the spherical coordinates for Schrödinger-like equations. The obtained estimates are sharp up to endpoints. The proof relies on estimates involving spherical averages, which were obtained in [5]. We discuss sharpness making use of a modified Knapp-type example.

Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition
Alexander Gladkov
2017, 16(6): 2053-2068 doi: 10.3934/cpaa.2017101 +[Abstract](909) +[HTML](2) +[PDF](413.9KB)
Abstract:

In this paper, we consider a semilinear parabolic equation with nonlinear nonlocal Neumann boundary condition and nonnegative initial datum. We first prove global existence result. We then give some criteria on this problem which determine whether the solutions blow up in finite time for large or for all nontrivial initial data. Finally, we show that under certain conditions blow-up occurs only on the boundary.

Multiplicity results for fractional systems crossing high eigenvalues
Fábio R. Pereira
2017, 16(6): 2069-2088 doi: 10.3934/cpaa.2017102 +[Abstract](661) +[HTML](2) +[PDF](462.8KB)
Abstract:

We investigate the existence of solutions for a system of nonlocal equations involving the fractional Laplacian operator and with nonlinearities reaching the subcritical growth and interacting, in some sense, with the spectrum of the operator.

Upper and lower time decay bounds for solutions of dissipative nonlinear Schrödinger equations
Nakao Hayashi, Chunhua Li and Pavel I. Naumkin
2017, 16(6): 2089-2104 doi: 10.3934/cpaa.2017103 +[Abstract](819) +[HTML](5) +[PDF](425.5KB)
Abstract:

We study the upper and lower time decay bounds for solutions of dissipative nonlinear Schrödinger equations

in space dimensions \begin{document} $n=1,2$ \end{document} or \begin{document} $3$ \end{document}, where \begin{document} $\lambda =\lambda _{1}+i\lambda _{2},$ \end{document} \begin{document} $\lambda _{j}∈ \mathbb{R},$ \end{document} \begin{document} $j=1,2,$ \end{document} \begin{document} $\lambda _{2}<0$ \end{document} and the subcritical order of nonlinearity \begin{document}$p=1+\frac{2}{n}-μ ,$\end{document} where \begin{document} $μ >0$ \end{document} is small enough.

Multiple solutions for a fractional nonlinear Schrödinger equation with local potential
Wulong Liu and Guowei Dai
2017, 16(6): 2105-2123 doi: 10.3934/cpaa.2017104 +[Abstract](790) +[HTML](13) +[PDF](497.3KB)
Abstract:

Using penalization techniques and the Ljusternik-Schnirelmann theory, we establish the multiplicity and concentration of solutions for the following fractional Schrödinger equation

where \begin{document} $0<α<1$ \end{document}, \begin{document} $N>2α$ \end{document}, \begin{document} $\varepsilon>0$ \end{document} is a small parameter, \begin{document} $V$ \end{document} satisfies the local condition, and \begin{document} $f$ \end{document} is superlinear and subcritical nonlinearity. We show that this equation has at least \begin{document} $\text{cat}_{M_{δ}}(M)$ \end{document} single spike solutions.

Existence of traveling waves for a class of nonlocal nonlinear equations with bell shaped kernels
Albert Erkip and Abba I. Ramadan
2017, 16(6): 2125-2132 doi: 10.3934/cpaa.2017105 +[Abstract](686) +[HTML](1) +[PDF](347.0KB)
Abstract:

In this article we are concerned with the existence of traveling wave solutions of a general class of nonlocal wave equations: \begin{document} $~u_{tt}-a^{2}u_{xx}=(β * u^{p})_{xx}$ \end{document}, \begin{document} $~p>1$ \end{document}. Members of the class arise as mathematical models for the propagation of waves in a wide variety of situations. We assume that the kernel \begin{document} $β $ \end{document} is a bell-shaped function satisfying some mild differentiability and growth conditions. Taking advantage of growth properties of bell-shaped functions, we give a simple proof for the existence of bell-shaped traveling wave solutions.

A Heteroclinic Solution to a Variational Problem Corresponding to FitzHugh-Nagumo type Reaction-Diffusion System with Heterogeneity
Takashi Kajiwara
2017, 16(6): 2133-2156 doi: 10.3934/cpaa.2017106 +[Abstract](732) +[HTML](5) +[PDF](443.0KB)
Abstract:

Chen, Kung and Morita [5] studied a variational problem corresponding to the FitzHugh-Nagumo type reaction-diffusion system (FHN type RD system), and they proved the existence of a heteroclinic solution to the system.

Motivated by [5], we consider a variational problem corresponding to FHN type RD system which involves heterogeneity. We prove the existence of a heteroclinic solution to the problem under certain conditions on the heterogeneity. Moreover, we give some information about the location of the transitions.

Multiple positive solutions for Kirchhoff type problems involving concave-convex nonlinearities
Jia-Feng Liao, Yang Pu, Xiao-Feng Ke and Chun-Lei Tang
2017, 16(6): 2157-2175 doi: 10.3934/cpaa.2017107 +[Abstract](761) +[HTML](5) +[PDF](431.3KB)
Abstract:

In this paper, we are interested in looking for multiple solutions for a class of Kirchhoff type problems with concave-convex nonlinearities. Under the combined effect of coefficient functions of concave-convex nonlinearities, by the Nehari method, we obtain two solutions, and one of them is a ground state solution. Under some stronger conditions, we point that the two solutions are positive solutions by the strong maximum principle.

Boundary Layer Problem and Quasineutral Limit of Compressible Euler-Poisson System
Shu Wang and Chundi Liu
2017, 16(6): 2177-2199 doi: 10.3934/cpaa.2017108 +[Abstract](916) +[HTML](1) +[PDF](459.7KB)
Abstract:

We study the boundary layer problem and the quasineutral limit of the compressible Euler-Poisson system arising from plasma physics in a domain with boundary. The quasineutral regime is the incompressible Euler equations. Compared to the quasineutral limit of compressible Euler-Poisson equations in whole space or periodic domain, the key difficulty here is to deal with the singularity caused by the boundary layer. The proof of the result is based on a λ-weighted energy method and the matched asymptotic expansion method.

Existence and convexity of solutions of the fractional heat equation
Antonio Greco and Antonio Iannizzotto
2017, 16(6): 2201-2226 doi: 10.3934/cpaa.2017109 +[Abstract](652) +[HTML](3) +[PDF](520.8KB)
Abstract:

We prove that the initial-value problem for the fractional heat equation admits an entire solution provided that the (possibly unbounded) initial datum has a conveniently moderate growth at infinity. Under the same growth condition we also prove that the solution is unique. The result does not require any sign assumption, thus complementing the Widder's type theorem of Barrios et al.[1] for positive solutions. Finally, we show that the fractional heat flow preserves convexity of the initial datum. Incidentally, several properties of stationary convex solutions are established.

Essential spectral singularities and the spectral expansion for the Hill operator
O. A. Veliev
2017, 16(6): 2227-2251 doi: 10.3934/cpaa.2017110 +[Abstract](799) +[HTML](14) +[PDF](447.3KB)
Abstract:

In this paper we investigate the spectral expansion for the one-dimensional Schrodinger operator with a periodic complex-valued potential. For this we consider in detail the spectral singularities and introduce new concepts as essential spectral singularities and singular quasimomenta.

Generalized Lorenz Equations for Acoustic-Gravity Waves in the Atmosphere. Attractors Dimension, Convergence and Homoclinic Trajectories
G. A. Leonov
2017, 16(6): 2253-2267 doi: 10.3934/cpaa.2017111 +[Abstract](656) +[HTML](1) +[PDF](2818.2KB)
Abstract:

Attractors dimension of Lorenz-Stenflo system is estimated. Convergence criteria are proved. Fishing principle for existence of homoclinic trajectory is applied.

The focusing NLS on exterior domains in three dimensions
Kai Yang
2017, 16(6): 2269-2297 doi: 10.3934/cpaa.2017112 +[Abstract](692) +[HTML](2) +[PDF](579.2KB)
Abstract:

We consider the Dirichlet problem of the focusing energy subcritical NLS outside a smooth compact strictly convex obstacle in dimension three. The critical space of our problem is \begin{document} $\dot{H}^s$ \end{document} with \begin{document} $0<s<1$ \end{document}. In this paper, we proved that if the initial data \begin{document} $u_{0}$ \end{document} satisfy \begin{document} $\Vert u_{0}\Vert _{2}^{1-s}\Vert \nabla u_{0}\Vert _{2}^{s}<\Vert \nabla Q\Vert _{2}^{s}\Vert Q\Vert _{2}^{1-s}$ \end{document} and \begin{document} $ M(u_{0})^{1-s}E(u_{0})^{s}<M(Q)^{1-s}E(Q)^{s},$ \end{document} then there exists a unique global solution which scatters in both time directions, where \begin{document} $Q$ \end{document} denotes the ground state solution in the whole space case.

Magnetic Laplacians of locally exact forms on the Sierpinski Gasket
Jessica Hyde, Daniel Kelleher, Jesse Moeller, Luke Rogers and Luis Seda
2017, 16(6): 2299-2319 doi: 10.3934/cpaa.2017113 +[Abstract](678) +[HTML](2) +[PDF](3373.4KB)
Abstract:

We give an explicit construction of a magnetic Schrödinger operator corresponding to a field with flux through finitely many holes of the Sierpinski Gasket. The operator is shown to have discrete spectrum accumulating at ∞, and it is shown that the asymptotic distribution of eigenvalues is the same as that for the Laplacian. Most eigenfunctions may be computed using gauge transformations corresponding to the magnetic field and the remainder of the spectrum may be approximated to arbitrary precision by using a sequence of approximations by magnetic operators on finite graphs.

Limit cycle bifurcations of a piecewise smooth Hamiltonian system with a generalized heteroclinic loop through a cusp
Jihua Yang, Erli Zhang and Mei Liu
2017, 16(6): 2321-2336 doi: 10.3934/cpaa.2017114 +[Abstract](805) +[HTML](2) +[PDF](618.1KB)
Abstract:

In this paper we study the limit cycle bifurcation of a piecewise smooth Hamiltonian system. By using the Melnikov function of piecewise smooth near-Hamiltonian systems, we obtain that at most \begin{document} $12n+7$ \end{document} limit cycles can bifurcate from the period annulus up to the first order in \begin{document} $\varepsilon$ \end{document}.

Dynamics of a Class of ODEs via Wavelets
Hildebrando M. Rodrigues, Tomás Caraballo and Marcio Gameiro
2017, 16(6): 2337-2355 doi: 10.3934/cpaa.2017115 +[Abstract](953) +[HTML](2) +[PDF](430.4KB)
Abstract:

The objective of this paper is to study a perturbed linear hyperbolic differential equation. The first part of this work is dedicated to study perturbation of the equilibrium (special solution) of a perturbed hyperbolic system. On the second part we analyze the stable and the unstable manifolds of a perturbed semilinear differential equation. We assume that the perturbed forcing function belongs to an \begin{document}$L_2$\end{document} class and that it is developed in a series of wavelets. Then we analyze the effect of this development on the special solution of the perturbed equation. Similar study is provided for the stable and unstable manifolds of this special solutions.

Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅰ: Dirichlet and Neumann boundary conditions
Anna Kostianko and Sergey Zelik
2017, 16(6): 2357-2376 doi: 10.3934/cpaa.2017116 +[Abstract](1004) +[HTML](3) +[PDF](442.9KB)
Abstract:

This is the first part of our study of inertial manifolds for the system of 1D reaction-diffusion-advection equations which is devoted to the case of Dirichlet or Neumann boundary conditions. Although this problem does not initially possess the spectral gap property, it is shown that this property is satisfied after the proper non-local change of the dependent variable. The case of periodic boundary conditions where the situation is principally different and the inertial manifold may not exist is considered in the second part of our study.

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