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Communications on Pure & Applied Analysis

September 2017 , Volume 16 , Issue 5

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Local well-posedness for 2-D Schrödinger equation on irrational tori and bounds on Sobolev norms
Seckin Demirbas
2017, 16(5): 1517-1530 doi: 10.3934/cpaa.2017072 +[Abstract](549) +[HTML](11) +[PDF](405.0KB)
Abstract:

In this paper we consider the cubic Schrödinger equation in two space dimensions on irrational tori. Our main result is an improvement of the Strichartz estimates on irrational tori. Using this estimate we obtain a local well-posedness result in \begin{document}$H^{s}$\end{document} for \begin{document}$s>\frac{131}{416} $\end{document}. We also obtain improved growth bounds for higher order Sobolev norms.

Existence of multiple positive weak solutions and estimates for extremal values for a class of concave-convex elliptic problems with an inverse-square potential
Yaoping Chen and Jianqing Chen
2017, 16(5): 1531-1552 doi: 10.3934/cpaa.2017073 +[Abstract](568) +[HTML](19) +[PDF](415.9KB)
Abstract:

In this paper, variational methods are used to establish some existence and multiplicity results and provide uniform estimates of extremal values for a class of elliptic equations of the form:

with Dirichlet boundary conditions, where \begin{document}$0∈ Ω\subset\mathbb{R}^N $\end{document}(\begin{document}$N≥q 3 $\end{document}) be a bounded domain with smooth boundary \begin{document}$\partial Ω $\end{document}, \begin{document}$μ>0 $\end{document} is a parameter, \begin{document}$0 < λ < Λ={{(N-2)^2}\over{4}}$, $0 < q < 1 < p < 2^*-1 $\end{document}, \begin{document}$h(x)>0 $\end{document} and \begin{document}$W(x) $\end{document} is a given function with the set \begin{document}$\{x∈ Ω: W(x)>0\} $\end{document} of positive measure.

On uniform estimate of complex elliptic equations on closed Hermitian manifolds
Wei Sun
2017, 16(5): 1553-1570 doi: 10.3934/cpaa.2017074 +[Abstract](673) +[HTML](31) +[PDF](411.7KB)
Abstract:

In this paper, we study Hessian equations and complex quotient equations on closed Hermitian manifolds. We directly derive the uniform estimate for the admissible solution. As an application, we solve general Hessian equations on closed Kähler manifolds.

Exponential boundary stabilization for nonlinear wave equations with localized damping and nonlinear boundary condition
Takeshi Taniguchi
2017, 16(5): 1571-1585 doi: 10.3934/cpaa.2017075 +[Abstract](549) +[HTML](14) +[PDF](402.6KB)
Abstract:

Let \begin{document}$ D\subset R^{d}$\end{document} be a bounded domain in the \begin{document}$d- $\end{document}dimensional Euclidian space \begin{document}$R^{d} $\end{document} with smooth boundary $Γ=\partial D.$ In this paper we consider exponential boundary stabilization for weak solutions to the wave equation with nonlinear boundary condition:

where \begin{document}$\left\| {{u_0}} \right\| < {\lambda _\beta }, $\end{document} \begin{document}$ E(0) < d_{β},$\end{document} where \begin{document}$λ_{β}, $\end{document} \begin{document}$d_{β} $\end{document} are defined in (21), (22) and \begin{document}$Γ=Γ_{0}\cupΓ_{1} $\end{document} and \begin{document}$\bar{Γ}_{0}\cap\bar{Γ}_{1}=φ. $\end{document}

Multiple positive solutions for Schrödinger-Poisson system in $\mathbb{R}^{3}$ involving concave-convex nonlinearities with critical exponent
Miao-Miao Li and Chun-Lei Tang
2017, 16(5): 1587-1602 doi: 10.3934/cpaa.2017076 +[Abstract](684) +[HTML](21) +[PDF](407.8KB)
Abstract:

In this paper, we study the existence of multiple positive solutions of the following Schrödinger-Poisson system with critical exponent

where \begin{document}$1 < q < 2 $\end{document} and \begin{document}$λ>0 $\end{document}. Under some appropriate conditions on \begin{document}$ l$\end{document} and \begin{document}$h $\end{document}, we show that there exists \begin{document}$λ^{*}>0 $\end{document} such that the above problem has at least two positive solutions for each \begin{document}$λ∈(0,λ^{*}) $\end{document} by using the Mountain Pass Theorem and Ekeland's Variational Principle.

Positive solutions for quasilinear Schrödinger equations in $\mathbb{R}^N$
Xiang-Dong Fang
2017, 16(5): 1603-1615 doi: 10.3934/cpaa.2017077 +[Abstract](670) +[HTML](24) +[PDF](414.8KB)
Abstract:

In this article we study the following quasilinear Schrödinger equation

where \begin{document}$ V(x)$\end{document} tends to some limit \begin{document}$V_{∞}>0 $\end{document} as \begin{document}$|x|\to∞ $\end{document} and \begin{document}$g∈ C(\mathbb{R},\mathbb{R}) $\end{document}. We prove the existence of positive solutions by using the Nehari manifold.

Gevrey regularity and existence of Navier-Stokes-Nernst-Planck-Poisson system in critical Besov spaces
Minghua Yang and Jinyi Sun
2017, 16(5): 1617-1639 doi: 10.3934/cpaa.2017078 +[Abstract](660) +[HTML](19) +[PDF](529.3KB)
Abstract:

The paper deals with the Cauchy problem of Navier-Stokes-Nernst-Planck-Poisson system (NSNPP). First of all, based on so-called Gevrey regularity estimates, which is motivated by the works of Foias and Temam [J. Funct. Anal., 87 (1989), 359-369], we prove that the solutions are analytic in a Gevrey class of functions. As a consequence of Gevrey estimates, we particularly obtain higher-order derivatives of solutions in Besov and Lebesgue spaces. Finally, we prove that there exists a positive constant \begin{document}$\mathbb{C}$\end{document} such that if the initial data \begin{document}$(u_{0}, n_{0}, c_{0})=(u_{0}^{h}, u_{0}^{3}, n_{0}, c_{0})$\end{document} satisfies

for \begin{document}$p, q, α$\end{document} with \begin{document}$1<p<q≤ 2p<\infty, \frac{1}{p}+\frac{1}{q}>\frac{1}{3}, 1< q<6, \frac{1}{p}-\frac{1}{q}≤\frac{1}{3}$\end{document}, then global existence of solutions with large initial vertical velocity component is established.

Existence and concentration for Kirchhoff type equations around topologically critical points of the potential
Yu Chen, Yanheng Ding and Suhong Li
2017, 16(5): 1641-1671 doi: 10.3934/cpaa.2017079 +[Abstract](689) +[HTML](21) +[PDF](578.5KB)
Abstract:

We consider the existence and concentration of solutions for the following Kirchhoff Type Equations

Under suitable conditions on the continuous functions \begin{document}$M$\end{document}, \begin{document}$V$\end{document} and \begin{document}$f$\end{document}, we obtain a family of positive solutions concentrating around the local maximum or saddle points of \begin{document}$V$\end{document}. Moreover with appropriate assumptions on \begin{document}$V$\end{document}, we also have multiple solutions clustering respectively around three kinds of critical points of \begin{document}$V$\end{document}.

Semilinear damped wave equation in locally uniform spaces
Martin Michálek, Dalibor Pražák and Jakub Slavík
2017, 16(5): 1673-1695 doi: 10.3934/cpaa.2017080 +[Abstract](732) +[HTML](13) +[PDF](480.7KB)
Abstract:

We study a damped wave equation with a nonlinear damping in the locally uniform spaces and prove well-posedness and existence of a locally compact attractor. An upper bound on the Kolmogorov's \begin{document}$\varepsilon$\end{document}-entropy is also established using the method of trajectories.

A new second critical exponent and life span for a quasilinear degenerate parabolic equation with weighted nonlocal sources
Lingwei Ma and Zhong Bo Fang
2017, 16(5): 1697-1706 doi: 10.3934/cpaa.2017081 +[Abstract](654) +[HTML](16) +[PDF](399.4KB)
Abstract:

In this paper, we consider positive solutions of a Cauchy problem for the following quasilinear degenerate parabolic equation with weighted nonlocal sources:

where \begin{document}$N≥1$\end{document}, \begin{document}$p>2$\end{document}, \begin{document}$q$\end{document}, \begin{document}$r≥1$\end{document}, \begin{document}$s≥0$\end{document}, and \begin{document}$r+s>1$\end{document}. We classify global and non-global solutions of the equation in the coexistence region by finding a new second critical exponent via the slow decay asymptotic behavior of an initial value at spatial infinity, and the life span of non-global solution is studied.

A direct method of moving planes for a fully nonlinear nonlocal system
Pengyan Wang and Pengcheng Niu
2017, 16(5): 1707-1718 doi: 10.3934/cpaa.2017082 +[Abstract](691) +[HTML](16) +[PDF](399.8KB)
Abstract:

In this paper we consider the system involving fully nonlinear nonlocal operators:

where \begin{document}$0<α, β<2, $\end{document} \begin{document}$p, q, r, s>1, $\end{document} \begin{document}$k_1(x), k_2(x)\geq0.$\end{document}

A narrow region principle and a decay at infinity are established for carrying on the method of moving planes. Then we prove the radial symmetry and monotonicity for positive solutions to the nonlinear system in the whole space. Furthermore non-existence of positive solutions to the system on a half space is derived.

Scale-free and quantitative unique continuation for infinite dimensional spectral subspaces of Schrödinger operators
Matthias Täufer and Martin Tautenhahn
2017, 16(5): 1719-1730 doi: 10.3934/cpaa.2017083 +[Abstract](714) +[HTML](16) +[PDF](422.8KB)
Abstract:

We prove a quantitative unique continuation principle for infinite dimensional spectral subspaces of Schrödinger operators. Let \begin{document}$Λ_L = (-L/2, L/2)^d$\end{document} and \begin{document}$H_L = -Δ_L + V_L$\end{document} be a Schrödinger operator on \begin{document}$L^2 (Λ_L)$\end{document} with a bounded potential \begin{document}$V_L : Λ_L \to \mathbb{R}^d$\end{document} and Dirichlet, Neumann, or periodic boundary conditions. Our main result is of the type

where \begin{document}$φ$\end{document} is an infinite complex linear combination of eigenfunctions of \begin{document}$H_L$\end{document} with exponentially decaying coefficients, \begin{document}$W_δ (L)$\end{document} is some union of equidistributed \begin{document}$δ$\end{document}-balls in \begin{document}$Λ_L$\end{document} and \begin{document}$C_{{\rm {sfuc}}} > 0$\end{document} an \begin{document}$L$\end{document}-independent constant. The exponential decay condition on \begin{document}$φ$\end{document} can alternatively be formulated as an exponential decay condition of the map \begin{document}$λ \mapsto \lVert χ_{[λ, ∞)} (H_L) φ \rVert^2$\end{document}. The novelty is that at the same time we allow the function \begin{document}$φ$\end{document} to be from an infinite dimensional spectral subspace and keep an explicit control over the constant \begin{document}$C_{{\rm {sfuc}}}$\end{document} in terms of the parameters. Moreover, we show that a similar result cannot hold under a polynomial decay condition.

Low Mach number limit of the full compressible Hall-MHD system
Jishan Fan, Fucai Li and Gen Nakamura
2017, 16(5): 1731-1740 doi: 10.3934/cpaa.2017084 +[Abstract](742) +[HTML](12) +[PDF](336.7KB)
Abstract:

In this paper we study the low Mach number limit of the full compressible Hall-magnetohydrodynamic (Hall-MHD) system in \begin{document}$\mathbb{T}^3$\end{document}. We prove that, as the Mach number tends to zero, the strong solution of the full compressible Hall-MHD system converges to that of the incompressible Hall-MHD system.

Semilinear nonlocal elliptic equations with critical and supercritical exponents
Mousomi Bhakta and Debangana Mukherjee
2017, 16(5): 1741-1766 doi: 10.3934/cpaa.2017085 +[Abstract](700) +[HTML](21) +[PDF](525.0KB)
Abstract:

We study the problem

where \begin{document}$s∈(0,1)$\end{document} is a fixed parameter, \begin{document}$(-Δ)^s$\end{document} is the fractional Laplacian in \begin{document}$\mathbb{R}^N$\end{document}, \begin{document}$q>p≥q \frac{N+2s}{N-2s}$\end{document} and \begin{document}$N>2s$\end{document}. For every \begin{document}$s∈(0,1)$\end{document}, we establish regularity results of solutions of above equation (whenever solution exists) and we show that every solution is a classical solution. Next, we derive certain decay estimate of solutions and the gradient of solutions at infinity for all \begin{document}$s∈(0,1)$\end{document}. Using those decay estimates, we prove Pohozaev type identity in ${{\mathbb{R}}^{N}}$ and we show that the above problem does not have any solution when \begin{document}$p=\frac{N+2s}{N-2s}$\end{document}. We also discuss radial symmetry and decreasing property of the solution and prove that when \begin{document}$p>\frac{N+2s}{N-2s}$\end{document}, the above problem admits a solution. Moreover, if we consider the above equation in a bounded domain with Dirichlet boundary condition, we prove that it admits a solution for every \begin{document}$p≥q \frac{N+2s}{N-2s}$\end{document} and every solution is a classical solution.

On some local-nonlocal elliptic equation involving nonlinear terms with exponential growth
Sami Aouaoui
2017, 16(5): 1767-1784 doi: 10.3934/cpaa.2017086 +[Abstract](471) +[HTML](19) +[PDF](413.2KB)
Abstract:

In this paper, we prove the existence of multiple solutions to some intermediate local-nonlocal elliptic equation in the whole two dimensional space. The nonlinearities exhibit an exponential growth at infinity.

Segregated vector Solutions for nonlinear Schrödinger systems with electromagnetic potentials
Jing Yang
2017, 16(5): 1785-1805 doi: 10.3934/cpaa.2017087 +[Abstract](600) +[HTML](16) +[PDF](470.3KB)
Abstract:

In this paper, we study the following nonlinear Schrödinger system in \begin{document}$\mathbb{R}^3$\end{document}

where \begin{document}$A(y)=A(|y|)∈ C^1(\mathbb{R}^3,\mathbb{R})$\end{document} is bounded, \begin{document}$λ_1(|y|),λ_2(|y|)$\end{document} are continuous positive radial potentials, and \begin{document}$β∈ \mathbb{R}$\end{document} is a coupling constant. We proved that if \begin{document}$A(y),λ_1(y),λ_2(y)$\end{document} satisfy some suitable conditions, the above problem has infinitely many non-radial segregated solutions.

Layered solutions to the vector Allen-Cahn equation in $\mathbb{R}^2$. Minimizers and heteroclinic connections
Giorgio Fusco
2017, 16(5): 1807-1841 doi: 10.3934/cpaa.2017088 +[Abstract](681) +[HTML](13) +[PDF](647.3KB)
Abstract:

Let \begin{document}$W:\mathbb{R}^m\to \mathbb{R}$\end{document} be a nonnegative potential with exactly two nondegenerate zeros \begin{document}$a_-≠ a_+∈\mathbb{R}^m$\end{document}. We assume that there are \begin{document}$N≥q 1$\end{document} distinct heteroclinic orbits connecting \begin{document}$a_-$\end{document} to \begin{document}$a_+$\end{document} represented by maps \begin{document}$\bar{u}_1,...,\bar{u}_N$\end{document} that minimize the one-dimensional energy \begin{document}$J_\mathbb{R}(u)=∈t_\mathbb{R}(\frac{\vert u^\prime\vert^2}{2}+W(u)){d} s$\end{document}.

We first consider the problem of characterizing the minimizers \begin{document}$u:\mathbb{R}^n\to \mathbb{R}^m$\end{document} of the energy \begin{document}$\mathcal{J}_Ω(u)=∈t_Ω(\frac{\vert\nabla u\vert^2}{2}+W(u)){d} x$\end{document}. Under a nondegeneracy condition on \begin{document}$\bar{u}_j$\end{document}, \begin{document}$j=1,...,N$\end{document} and in two space dimensions, we prove that, provided it remains away from \begin{document}$a_-$\end{document} and \begin{document}$a_+$\end{document} in corresponding half spaces \begin{document}$S_-$\end{document} and \begin{document}$S_+$\end{document}, a bounded minimizer \begin{document}$u:\mathbb{R}^2\to \mathbb{R}^m$\end{document} is necessarily an heteroclinic connection between suitable translates \begin{document}$\bar{u}_-(·-η_-)$\end{document} and \begin{document}$\bar{u}_+(·-η_+)$\end{document} of some \begin{document}$\bar{u}_±∈\{\bar{u}_1,...,\bar{u}_N\}$\end{document}.

Then we focus on the existence problem and assuming \begin{document}$N=2$\end{document} and denoting \begin{document}$\bar{u}_-,\bar{u}_+$\end{document} the representations of the two orbits connecting \begin{document}$a_-$\end{document} to \begin{document}$a_+$\end{document} we give a new proof of the existence (first proved in [32]) of a solution \begin{document}$u:\mathbb{R}^2\to \mathbb{R}^m$\end{document} of

that connects certain translates of \begin{document}$\bar{u}_±$\end{document}.

Optimality conditions of the first eigenvalue of a fourth order Steklov problem
Monika Laskawy
2017, 16(5): 1843-1859 doi: 10.3934/cpaa.2017089 +[Abstract](598) +[HTML](15) +[PDF](365.0KB)
Abstract:

In this paper we compute the first and second general domain variation of the first eigenvalue of a fourth order Steklov problem. We study optimality conditions for the ball among domains of given measure and among domains of given perimeter. We show that in both cases the ball is a local minimizer among all domains of equal measure and perimeter.

Global well-posedness of the two-dimensional horizontally filtered simplified Bardina turbulence model on a strip-like region
Luca Bisconti and Davide Catania
2017, 16(5): 1861-1881 doi: 10.3934/cpaa.2017090 +[Abstract](519) +[HTML](15) +[PDF](440.7KB)
Abstract:

We consider the 2D simplified Bardina turbulence model, with horizontal filtering, in an unbounded strip-like domain. We prove global existence and uniqueness of weak solutions in a suitable class of anisotropic weighted Sobolev spaces.

Global dynamics of a microorganism flocculation model with time delay
Songbai Guo and Wanbiao Ma
2017, 16(5): 1883-1891 doi: 10.3934/cpaa.2017091 +[Abstract](586) +[HTML](18) +[PDF](404.7KB)
Abstract:

In this paper, we consider a microorganism flocculation model with time delay. In this model, there may exist a forward bifurcation/backward bifurcation. By constructing suitable positively invariant sets and using Lyapunov-LaSalle theorem, we study the global stability of the equilibria of the model under certain conditions. Furthermore, we also investigate the permanence of the model, and an explicit expression of the eventual lower bound of microorganism concentration is given.

Dynamics of some stochastic chemostat models with multiplicative noise
T. Caraballo, M. J. Garrido-Atienza and J. López-de-la-Cruz
2017, 16(5): 1893-1914 doi: 10.3934/cpaa.2017092 +[Abstract](704) +[HTML](31) +[PDF](6682.6KB)
Abstract:

In this paper we study two stochastic chemostat models, with and without wall growth, driven by a white noise. Specifically, we analyze the existence and uniqueness of solutions for these models, as well as the existence of the random attractor associated to the random dynamical system generated by the solution. The analysis will be carried out by means of the well-known Ornstein-Uhlenbeck process, that allows us to transform our stochastic chemostat models into random ones.

Structure-preserving finite difference schemes for the Cahn-Hilliard equation with dynamic boundary conditions in the one-dimensional case
Takeshi Fukao, Shuji Yoshikawa and Saori Wada
2017, 16(5): 1915-1938 doi: 10.3934/cpaa.2017093 +[Abstract](622) +[HTML](16) +[PDF](688.0KB)
Abstract:

The structure-preserving finite difference schemes for the one dimensional Cahn-Hilliard equation with dynamic boundary conditions are studied. A dynamic boundary condition is a sort of transmission condition that includes the time derivative, namely, it is itself a time evolution equation. The Cahn-Hilliard equation with dynamic boundary conditions is well-treated from various viewpoints. The standard type consists of a dynamic boundary condition for the order parameter, and the Neumann boundary condition for the chemical potential. Recently, Goldstein-Miranville-Schimperna proposed a new type of dynamic boundary condition for the Cahn-Hilliard equation. In this article, numerical schemes for the problem with these two kinds of dynamic boundary conditions are introduced. In addition, a mathematical result on the existence of a solution for the scheme with an error estimate is also obtained for the former boundary condition.

Corrigendum to "On small data scattering of Hartree equations with short-range interaction" [Comm. Pure. Appl. Anal., 15 (2016), 1809-1823]
Yonggeun Cho, Gyeongha Hwang and Tohru Ozawa
2017, 16(5): 1939-1940 doi: 10.3934/cpaa.2017094 +[Abstract](478) +[HTML](16) +[PDF](271.9KB)
Abstract:

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