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Communications on Pure & Applied Analysis

July 2020 , Volume 19 , Issue 7

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The Lazer-McKenna conjecture for an anisotropic planar elliptic problem with exponential Neumann data
Yibin Zhang
2020, 19(7): 3445-3476 doi: 10.3934/cpaa.2020151 +[Abstract](220) +[HTML](67) +[PDF](576.06KB)
Abstract:

Let \begin{document}$ \Omega\subset\mathbb{R}^2 $\end{document} be a bounded smooth domain, we study the following anisotropic elliptic problem

where \begin{document}$ \nu $\end{document} denotes the outer unit normal vector to \begin{document}$ \partial\Omega $\end{document}, \begin{document}$ h\in C^{0, \alpha}( \partial\Omega) $\end{document}, \begin{document}$ s>0 $\end{document} is a large parameter, \begin{document}$ a(x) $\end{document} is a positive smooth function and \begin{document}$ \phi_1 $\end{document} is a positive first Steklov eigenfunction. We show that this problem has an unbounded number of solutions for all sufficiently large \begin{document}$ s $\end{document}, which give a positive answer to a generalization of the Lazer-McKenna conjecture for this case. Moreover, the solutions found exhibit multiple concentration behavior around boundary maxima of \begin{document}$ a(x)\phi_1 $\end{document} as \begin{document}$ s\rightarrow+\infty $\end{document}.

Generalizations of $ p $-Laplace operator for image enhancement: Part 2
George Baravdish, Yuanji Cheng, Olof Svensson and Freddie Åström
2020, 19(7): 3477-3500 doi: 10.3934/cpaa.2020152 +[Abstract](162) +[HTML](86) +[PDF](1391.03KB)
Abstract:

We have in a previous study introduced a novel elliptic operator \begin{document}$ \Delta_{(p, q)} u = |\nabla u|^q\Delta_1 u +(p-1)|\nabla u|^{p-2} \Delta_{\infty} u $\end{document}, \begin{document}$ p \ge 1 $\end{document}, \begin{document}$ q\ge 0, $\end{document} as a generalization of the \begin{document}$ p $\end{document}-Laplace operator. In this paper, we establish the well-posedness of the parabolic equation \begin{document}$ u_{t} = |\nabla u|^{1-q}\Delta_{(1+q, q)}, $\end{document} where \begin{document}$ q = q(|\nabla u|) $\end{document} is continuous and has range in \begin{document}$ [0, 1], $\end{document} in the framework of viscosity solutions. We prove the consistency and convergence of the numerical scheme of finite differences of this parabolic equation. Numerical simulations shows the advantage of this operator applied to image enhancement.

Multiplicity of radial and nonradial solutions to equations with fractional operators
Norihisa Ikoma
2020, 19(7): 3501-3530 doi: 10.3934/cpaa.2020153 +[Abstract](197) +[HTML](64) +[PDF](588.67KB)
Abstract:

In this paper, we study the existence of radial and nonradial solutions to the scalar field equations with fractional operators. For radial solutions, we prove the existence of infinitely many solutions under \begin{document}$ N \geq 2 $\end{document}. We also show the existence of least energy solution (with the Pohozaev identity) and its mountain pass characterization. For nonradial solutions, we prove the existence of at least one nonradial solution under \begin{document}$ N \geq 4 $\end{document} and infinitely many nonradial solutions under either \begin{document}$ N = 4 $\end{document} or \begin{document}$ N \geq 6 $\end{document}. We treat both of the zero mass and the positive mass cases.

General decay for a viscoelastic rotating Euler-Bernoulli beam
Ammar Khemmoudj and Imane Djaidja
2020, 19(7): 3531-3557 doi: 10.3934/cpaa.2020154 +[Abstract](327) +[HTML](80) +[PDF](517.91KB)
Abstract:

In this paper, we consider a viscoelastic rotating Euler-Bernoulli beam that has one end fixed to a rotated motor in a horizontal plane and to a tip mass at the other end. For a large class relaxation function \begin{document}$ q $\end{document}, namely, \begin{document}$ q^{\prime}(t) \leq -\zeta(t)H(q(t)) $\end{document}, where \begin{document}$ H $\end{document} is an increasing and convex function near the origin and \begin{document}$ \zeta $\end{document} is a nonincreasing function, we establish optimal explicit and general energy decay results from which we can recover the optimal exponential and polynomial decay.

The Hardy–Moser–Trudinger inequality via the transplantation of Green functions
Van Hoang Nguyen
2020, 19(7): 3559-3574 doi: 10.3934/cpaa.2020155 +[Abstract](204) +[HTML](79) +[PDF](426.15KB)
Abstract:

We provide a new proof of the Hardy–Moser–Trudinger inequality and the existence of its extremals which are established by Wang and Ye ("G. Wang, and D. Ye, A Hardy–Moser–Trudinger inequality, Adv. Math, 230 (2012) 294–230.") without using the blow-up analysis method. Our proof is based on the transformation of functions via the transplantation of Green functions. This method enables us to compute explicitly the concentrating level of the Hardy–Moser–Trudinger functional over the normalizing concentrating sequences which is crucial to prove the existence of extremals for the Hardy–Moser–Trudinger inequality. Some comments on the applications of this approach to some other Moser–Trudinger type inequalities are given.

General condition for exponential stability of thermoelastic Bresse systems with Cattaneo's law
Pedro Roberto de Lima and Hugo D. Fernández Sare
2020, 19(7): 3575-3596 doi: 10.3934/cpaa.2020156 +[Abstract](138) +[HTML](107) +[PDF](494.25KB)
Abstract:

In this paper, we give a new and more general sufficient condition for exponential stability of thermoelastic Bresse systems with heat flux given by Cattaneo's law acting in shear and longitudinal motion equations. This condition, which we also prove to be necessary in some special cases, is given by a relation between the constants of the system and generalizes the well-known equal wave speed condition.

Property of solutions for elliptic equation involving the higher-order fractional Laplacian in $ \mathbb{R}^n_+ $
Mei Yu, Xia Zhang and Binlin Zhang
2020, 19(7): 3597-3612 doi: 10.3934/cpaa.2020157 +[Abstract](128) +[HTML](62) +[PDF](419.97KB)
Abstract:

In this paper, we consider the following equation with the higher-order fractional Laplacian \begin{document}$ (-\Delta)^s $\end{document} for \begin{document}$ s = m+\frac{\alpha}{2} $\end{document}:

where \begin{document}$ m\in \mathbb{N}^* $\end{document}, \begin{document}$ 0<\alpha<2 $\end{document}. By developing a narrow region principle in unbounded domain and establishing a equivalence of differential equation and integral equation, together with the method of moving planes, we deduce the monotonicity property of positive solutions and the Liouville theorem of nonnegative solutions.

A lower bound for the principal eigenvalue of fully nonlinear elliptic operators
Pablo Blanc
2020, 19(7): 3613-3623 doi: 10.3934/cpaa.2020158 +[Abstract](114) +[HTML](52) +[PDF](950.13KB)
Abstract:

In this article we present a new technique to obtain a lower bound for the principal Dirichlet eigenvalue of a fully nonlinear elliptic operator. We illustrate the construction of an appropriate radial function required to obtain the bound in several examples. In particular we use our results to prove that

where \begin{document}$ R $\end{document} is the largest radius of a ball included in the domain \begin{document}$ \Omega\subset {\mathbb R}^n $\end{document}, and \begin{document}$ \lambda_{1,p}(\Omega) $\end{document} and \begin{document}$ \lambda_{1,\infty}(\Omega) $\end{document} are the principal eigenvalue for the homogeneous \begin{document}$ p $\end{document}-laplacian and the homogeneous infinity laplacian respectively.

The Cauchy problem for heat equation with fractional Laplacian and exponential nonlinearity
Ahmad Z. Fino and Mokhtar Kirane
2020, 19(7): 3625-3650 doi: 10.3934/cpaa.2020160 +[Abstract](140) +[HTML](60) +[PDF](506.04KB)
Abstract:

We consider the Cauchy problem for heat equation with fractional Laplacian and exponential nonlinearity. We establish local well-posedness result in Orlicz spaces. We derive the existence of global solutions for small initial data. We obtain decay estimates for large time in Lebesgue spaces.

The dynamics of nonlocal diffusion systems with different free boundaries
Lei Li, Jianping Wang and Mingxin Wang
2020, 19(7): 3651-3672 doi: 10.3934/cpaa.2020161 +[Abstract](164) +[HTML](63) +[PDF](451.76KB)
Abstract:

This paper is concerned with a class of free boundary models with "nonlocal diffusions'' and different free boundaries, which are natural extensions of free boundary problems of reaction diffusion systems with different free boundaries in [M.X.Wang and Y.Zhang, J. Differ. Equ., 264 (2018), 3527-3558] and references therein. These different free boundaries, which may intersect each other as time evolves, are used to describe the spreading front of the species. We prove that such kind of nonlocal diffusion problems has a unique global solution. Moreover, we investigate the long time behavior of global solution and criteria of spreading and vanishing for the classical Lotka-Volterra competition, prey-predator and mutualist models.

Improved Sobolev inequalities and critical problems
Xiaoli Chen and Jianfu Yang
2020, 19(7): 3673-3695 doi: 10.3934/cpaa.2020162 +[Abstract](154) +[HTML](73) +[PDF](477.42KB)
Abstract:

In this paper, we establish two refinement of Sobolev-Hardy inequalities in terms of Morrey spaces. Then, with help of these inequalities, we show the existence of nontrivial solutions for doubly critical problems in \begin{document}$ \mathbb{R}^N $\end{document} involving \begin{document}$ p $\end{document}-Laplacian

where \begin{document}$ x = (y,z)\in\mathbb{R}^K\times\mathbb{R}^{N-K},1\leq K\leq N,1<p<N,0<\alpha,\beta<\min\{K,\frac{NKp}{N^2-(N-K)p}\} $\end{document} and \begin{document}$ p^\ast_{\alpha} = \frac{p(N-\alpha)}{N-p} $\end{document} is the critical Hardy-Sobolev exponent, and critical problems involving fractional Laplacian

where \begin{document}$ 0<s<\frac{N}2,0<\alpha,\beta<\min\{K,\frac{2NKs}{N^2-2s(N-K)}\} $\end{document} and \begin{document}$ 2^\ast_{s,\alpha} = \frac{2(N-\alpha)}{N-2s} $\end{document}.

Bound state positive solutions for a class of elliptic system with Hartree nonlinearity
Guofeng Che, Haibo Chen and Tsung-fang Wu
2020, 19(7): 3697-3722 doi: 10.3934/cpaa.2020163 +[Abstract](132) +[HTML](75) +[PDF](535.36KB)
Abstract:

In this paper, we are concerned with the following two-component system of Schrödinger equations with Hartree nonlinearity:

where \begin{document}$ 0<\varepsilon \ll 1 $\end{document} is a small parameter, \begin{document}$ 0<q\leq p $\end{document}, \begin{document}$ I_{\varepsilon}(x) = \frac{\Gamma((N-\alpha)/2)} {\Gamma(\alpha/2)\pi^{\frac{N}{2}}2^{\alpha}\varepsilon^{\alpha}}\frac{1}{|x|^{N-\alpha}}, \; x\in\mathbb{R}^{N}\setminus\{0\} $\end{document}, \begin{document}$ \alpha\in(0,N),\; N = 3,4,5 $\end{document} and \begin{document}$ \lambda _{l}\geq0,\; \mu _{l}>0,\; l = 1,2, $\end{document} are constants. Under some suitable assumptions on the potentials \begin{document}$ V_{l}(x),l = 1,2, $\end{document} and the coupled function \begin{document}$ \beta(x) $\end{document}, we prove the existence and multiplicity of positive solutions for the above system by using energy estimates, the Nehari manifold technique and the Lusternik-Schnirelmann theory. Furthermore, the existence and nonexistence of the least energy positive solutions are also explored.

Symmetry of positive solutions to fractional equations in bounded domains and unbounded cylinders
Yunyun Hu
2020, 19(7): 3723-3734 doi: 10.3934/cpaa.2020164 +[Abstract](107) +[HTML](72) +[PDF](378.2KB)
Abstract:

The aim of this paper is to study symmetry and monotonicity for positive solutions to fractional equations. We first consider the following problems in bounded domains in the sense of distributions

where \begin{document}$ n>2s $\end{document}, \begin{document}$ 0<s<1 $\end{document}. We prove that all positive solutions are radically symmetric about the origin. Compare to results in [1], we use a completely different method under the weaker conditions in \begin{document}$ f $\end{document}. Next we consider a problem in infinite cylinders. We establish the symmetry and monotonicity of positive solutions by using the method of moving planes. This result can be seen as the nonlocal counterparts of [3].

Positive bound states for fractional Schrödinger-Poisson system with critical exponent
Xia Sun and Kaimin Teng
2020, 19(7): 3735-3768 doi: 10.3934/cpaa.2020165 +[Abstract](161) +[HTML](93) +[PDF](599.29KB)
Abstract:

In this paper, we consider the following fractional critical Schrödinger-Poisson system without perturbation terms

where \begin{document}$ s\in(\frac{3}{4},1) $\end{document}, \begin{document}$ t\in(0,1) $\end{document}. Under some suitable assumptions on potentials \begin{document}$ V(x) $\end{document} and \begin{document}$ K(x) $\end{document}, we show that the ground state positive solutions do not exist, by using the topological argument, we establish the existence of positive bound state solutions in the range \begin{document}$ (\frac{s}{3}\mathcal{S}_s^{\frac{3}{2s}},\frac{2s}{3}\mathcal{S}_s^{\frac{3}{2s}}) $\end{document}.

Blow-up for two-component Camassa-Holm equation with generalized weak dissipation
Wenxia Chen, Jingyi Liu, Danping Ding and Lixin Tian
2020, 19(7): 3769-3784 doi: 10.3934/cpaa.2020166 +[Abstract](130) +[HTML](54) +[PDF](384.33KB)
Abstract:

This paper is concerned with blow-up solution for the Cauchy problem of two-component Camassa-Holm equation with generalized weak dissipation. By Kato's theorem and monotonicity, we investigate the local well-posedness of Cauchy problem and establish the blow-up criteria and the blow-up rate. Moreover, the property of blow-up points set is characterized.

The probabilistic Cauchy problem for the fourth order Schrödinger equation with special derivative nonlinearities
Shuai Zhang and Shaopeng Xu
2020, 19(7): 3785-3803 doi: 10.3934/cpaa.2020167 +[Abstract](126) +[HTML](54) +[PDF](551.59KB)
Abstract:

Chen and Zhang [7] consider the probabilistic Cauchy problem of the fourth order Schrödinger equation

where \begin{document}$ P_m $\end{document} is a homogeneous polynomial of degree \begin{document}$ m $\end{document}. The almost sure local well-posedness and small data global existence were obtained in \begin{document}$ H^s(\mathbb{R}^d) $\end{document} with the regularity threshold \begin{document}$ s_c-1/2 $\end{document} when \begin{document}$ d\geq3 $\end{document}, where \begin{document}$ s_c: = d/2-2/(m-1) $\end{document} is the scaling critical regularity. For the lower regularity threshold \begin{document}$ (d-1)s_c/d $\end{document} with \begin{document}$ m = 2 $\end{document} and \begin{document}$ s_c-\min\{1, d/4\} $\end{document} with \begin{document}$ m\geq3 $\end{document}, we get the corresponding well-posedness of the following fourth order nonlinear Schrödinger equation

on \begin{document}$ {\mathbb{R}}^d $\end{document} (\begin{document}$ d\geq2 $\end{document}) with random initial data.

Approximation of the trajectory attractor of the 3D smectic-A liquid crystal flow equations
Xiuqing Wang, Yuming Qin and Alain Miranville
2020, 19(7): 3805-3827 doi: 10.3934/cpaa.2020168 +[Abstract](130) +[HTML](58) +[PDF](497.14KB)
Abstract:

In this paper, we first establish the existence of trajectory attractors for the 3D smectic-A liquid crystal flow system and 3D smectic-A liquid crystal flow-\begin{document}$ \alpha $\end{document} model, and then prove that the latter trajectory attractor converges to the former one as the parameter \begin{document}$ \alpha\rightarrow 0^{+} $\end{document}.

Algebraic structure of the $ L_2 $ analytic Fourier–Feynman transform associated with Gaussian paths on Wiener space
Jae Gil Choi and David Skoug
2020, 19(7): 3829-3842 doi: 10.3934/cpaa.2020169 +[Abstract](111) +[HTML](56) +[PDF](438.54KB)
Abstract:

In this paper we study algebraic structures of the classes of the \begin{document}$ L_2 $\end{document} analytic Fourier–Feynman transforms on Wiener space. To do this we first develop several rotation properties of the generalized Wiener integral associated with Gaussian paths. We then proceed to analyze the \begin{document}$ L_2 $\end{document} analytic Fourier–Feynman transforms associated with Gaussian paths. Our results show that these \begin{document}$ L_2 $\end{document} analytic Fourier–Feynman transforms are actually linear operator isomorphisms from a Hilbert space into itself. We finally investigate the algebraic structures of these classes of the transforms on Wiener space, and show that they indeed are group isomorphic.

Global boundedness of solutions to a chemotaxis-fluid system with singular sensitivity and logistic source
Guoqiang Ren and Bin Liu
2020, 19(7): 3843-3883 doi: 10.3934/cpaa.2020170 +[Abstract](147) +[HTML](68) +[PDF](662.91KB)
Abstract:

In this paper, we investigate the chemotaxis-fluid system with singular sensitivity and logistic source in bounded convex domain with smooth boundary. We present the global existence of very weak solutions under appropriate regularity assumptions on the initial data. Then, we show that system possesses a global bounded classical solution. Finally, we present a unique globally bounded classical solution for a fluid-free system. In addition, the asymptotic behavior of the solutions is studied, and our results generalize and improve some well-known results in the literature, and partially results are new.

Blow-up for semilinear wave equations with time-dependent damping in an exterior domain
Mohamed Jleli and Bessem Samet
2020, 19(7): 3885-3900 doi: 10.3934/cpaa.2020143 +[Abstract](183) +[HTML](160) +[PDF](418.51KB)
Abstract:

We consider the semilinear wave equation with time-dependent damping

where \begin{document}$ D^c = \mathbb{R}^N\backslash D $\end{document}, \begin{document}$ D $\end{document} is the closed unit ball in \begin{document}$ \mathbb{R}^N $\end{document}, \begin{document}$ N\geq 2 $\end{document}, \begin{document}$ \mu>0 $\end{document}, \begin{document}$ p>1 $\end{document} and \begin{document}$ -1<\beta<1 $\end{document}. The considered equation is investigated under the boundary conditions:

where \begin{document}$ n^+ $\end{document} is the outward (relative to \begin{document}$ D^c $\end{document}) unit normal on \begin{document}$ \partial D $\end{document}. General blow-up results are established for the considered problems. Moreover, for a certain class of functions \begin{document}$ b $\end{document}, the critical exponent in the sense of Fujita is obtained.

A trace theorem for Sobolev spaces on the Sierpinski gasket
Shiping Cao, Shuangping Li, Robert S. Strichartz and Prem Talwai
2020, 19(7): 3901-3916 doi: 10.3934/cpaa.2020159 +[Abstract](122) +[HTML](49) +[PDF](434.92KB)
Abstract:

We give a discrete characterization of the trace of a class of Sobolev spaces on the Sierpinski gasket to the bottom line. This includes the \begin{document}$ L^2 $\end{document} domain of the Laplacian as a special case. In addition, for Sobolev spaces of low orders, including the domain of the Dirichlet form, the trace spaces are Besov spaces on the line.

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