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Communications on Pure & Applied Analysis

September 2018 , Volume 17 , Issue 5

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Homoclinic solutions of discrete $ \phi $-Laplacian equations with mixed nonlinearities
Genghong Lin and Zhan Zhou
2018, 17(5): 1723-1747 doi: 10.3934/cpaa.2018082 +[Abstract](513) +[HTML](214) +[PDF](490.86KB)
Abstract:

By using critical point theory, we obtain some new sufficient conditions on the existence of homoclinic solutions of a class of nonlinear discrete \begin{document}$ \phi $\end{document}-Laplacian equations with mixed nonlinearities for the potentials being periodic or being unbounded, respectively. And we prove it is also necessary in some special cases. In addition, multiplicity results of homoclinic solutions for nonlinear discrete \begin{document}$ \phi $\end{document}-Laplacian equations with unbounded potentials have also been considered. In our paper, the nonlinearities can be mixed super \begin{document}$ p $\end{document}-linear with asymptotically \begin{document}$ p $\end{document}-linear at \begin{document}$ ∞ $\end{document} for \begin{document}$ p≥ 1 $\end{document}. To the best of our knowledge, there is no such result for the existence of homoclinic solutions with discrete \begin{document}$ \phi $\end{document}-Laplacian before. Finally, an extension has also been considered.

On existence and nonexistence of positive solutions of an elliptic system with coupled terms
Yayun Li and Yutian Lei
2018, 17(5): 1749-1764 doi: 10.3934/cpaa.2018083 +[Abstract](572) +[HTML](195) +[PDF](451.06KB)
Abstract:

This paper is concerned with the elliptic system

where \begin{document}$ n ≥ 3 $\end{document}, \begin{document}$ p,q>0 $\end{document} and \begin{document}$ \max\{p,q\} ≥ 1 $\end{document}. We discuss the nonexistence of positive solutions in subcritical case and stable solutions in supercritical case, the necessary and sufficient conditions of classification in the critical case, and the Joseph-Lundgren-type condition for existence of local stable solutions.

Positive radial solutions of a nonlinear boundary value problem
Patricio Cerda, Leonelo Iturriaga, Sebastián Lorca and Pedro Ubilla
2018, 17(5): 1765-1783 doi: 10.3934/cpaa.2018084 +[Abstract](512) +[HTML](139) +[PDF](420.57KB)
Abstract:

In this work we study the following quasilinear elliptic equation:

where \begin{document}$ a $\end{document} is a positive continuous function, \begin{document}$ g $\end{document} is a nonnegative and nondecreasing continuous function, \begin{document}$ Ω = B_R $\end{document}, is the ball of radius \begin{document}$ R>0 $\end{document} centered at the origin in \begin{document}$ \mathbb{R} ^N $\end{document}, \begin{document}$N≥3 $\end{document} and, the constants \begin{document}$ α,β∈\mathbb{R} $\end{document}, \begin{document}$ γ∈(0,1) $\end{document} and \begin{document}$ p>1 $\end{document}.

We derive a new Liouville type result for a kind of "broken equation". This result together with blow-up techniques, a priori estimates and a fixed-point result of Krasnosel'skii, allow us to ensure the existence of a positive radial solution. In this paper we also obtain a non-existence result, proven through a variation of the Pohozaev identity.

On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities
Binhua Feng
2018, 17(5): 1785-1804 doi: 10.3934/cpaa.2018085 +[Abstract](468) +[HTML](155) +[PDF](389.1KB)
Abstract:

This paper is devoted to the analysis of blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities

where \begin{document}$ 0<p_1<p_2<\frac{2s}{N-2s}$ \end{document}. Firstly, we obtain some sufficient conditions about existence of blow-up solutions, and then derive some sharp thresholds of blow-up and global existence by constructing some new estimates. Moreover, we find the sharp threshold mass of blow-up and global existence in the case \begin{document}$ 0<p_1<\frac{2s}{N}$ \end{document} and \begin{document} $p_2 = \frac{2s}{N}$ \end{document}. Finally, we investigate the dynamical properties of blow-up solutions, including \begin{document} $L^2$\end{document}-concentration, blow-up rate and limiting profile.

Global existence and blow-up of solutions to a singular Non-Newton polytropic filtration equation with critical and supercritical initial energy
Guangyu Xu and Jun Zhou
2018, 17(5): 1805-1820 doi: 10.3934/cpaa.2018086 +[Abstract](342) +[HTML](148) +[PDF](407.28KB)
Abstract:

In this paper, we revisit the singular Non-Newton polytropic filtration equation, which was studied extensively in the recent years. However, all the studies are mostly concerned with subcritical initial energy, i.e., \begin{document} $E(u_0)<d$ \end{document}, where \begin{document} $E(u_0)$ \end{document} is the initial energy and \begin{document} $d$ \end{document} is the mountain-pass level. The main purpose of this paper is to study the behaviors of the solution with \begin{document} $E(u_0)≥d$ \end{document} by potential well method and some differential inequality techniques.

A free boundary problem for the Fisher-KPP equation with a given moving boundary
Hiroshi Matsuzawa
2018, 17(5): 1821-1852 doi: 10.3934/cpaa.2018087 +[Abstract](461) +[HTML](155) +[PDF](437.41KB)
Abstract:

We study free boundary problem of Fisher-KPP equation \begin{document} $u_t = u_{xx}+u(1-u), \ t>0, \ ct<x<h(t)$ \end{document}. The number \begin{document} $c>0$ \end{document} is a given constant, \begin{document} $h(t)$ \end{document} is a free boundary which is determined by the Stefan-like condition. This model may be used to describe the spreading of a non-native species over a one dimensional habitat. The free boundary \begin{document} $x = h(t)$ \end{document} represents the spreading front. In this model, we impose zero Dirichlet condition at left moving boundary \begin{document} $x = ct$ \end{document}. This means that the left boundary of the habitat is a very hostile environment and that the habitat is eroded away by the left moving boundary at constant speed \begin{document} $c$ \end{document}.

In this paper we will give a trichotomy result, that is, for any initial data, exactly one of the three behaviors, vanishing, spreading and transition, happens. This result is related to the results appear in the free boundary problem for the Fisher-KPP equation with a shifting-environment, which was considered by Du, Wei and Zhou [11]. However the vanishing in our problem is different from that in [11] because in our vanishing case, the solution is not global-in-time.

Global Carleman estimate for the Kawahara equation and its applications
Peng Gao
2018, 17(5): 1853-1874 doi: 10.3934/cpaa.2018088 +[Abstract](343) +[HTML](184) +[PDF](406.43KB)
Abstract:

In this paper, we establish a global Carleman estimate for the Kawahara equation. Based on this estimate, we obtain the Unique Continuation Property (UCP) for this equation and the global exponential stability for the Kawahara equation with a very weak localized dissipation.

Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential
Helin Guo, Yimin Zhang and Huansong Zhou
2018, 17(5): 1875-1897 doi: 10.3934/cpaa.2018089 +[Abstract](430) +[HTML](156) +[PDF](411.35KB)
Abstract:

We study a Kirchhoff type elliptic equation with trapping potential. The existence and blow-up behavior of solutions with normalized \begin{document}$L^{2}$\end{document}-norm for this equation are discussed.

Existence and asymptotic behaviors of traveling waves of a modified vector-disease model
Zengji Du and Zhaosheng Feng
2018, 17(5): 1899-1920 doi: 10.3934/cpaa.2018090 +[Abstract](350) +[HTML](150) +[PDF](365.31KB)
Abstract:

In this paper, we are concerned with the existence and asymptotic behavior of traveling wave fronts in a modified vector-disease model. We establish the existence of traveling wave solutions for the modified vector-disease model without delay, then explore the existence of traveling fronts for the model with a special local delay convolution kernel by employing the geometric singular perturbation theory and the linear chain trick. Finally, we deal with the local stability of the steady states, the existence and asymptotic behaviors of traveling wave solutions for the model with the convolution kernel of a special non-local delay.

Global synchronising behavior of evolution equations with exponentially growing nonautonomous forcing
Xuewei Ju and Desheng Li
2018, 17(5): 1921-1944 doi: 10.3934/cpaa.2018091 +[Abstract](349) +[HTML](152) +[PDF](406.9KB)
Abstract:

This work is concerned with the following nonautonomous evolutionary system on a Banach space \begin{document}$X$\end{document},

where \begin{document}$A$\end{document} is a hyperbolic sectorial operator on \begin{document}$X$\end{document}, the nonlinearity \begin{document}$f \in C({X^\alpha } \times X,X)$\end{document} is Lipschitz in the first variable, the nonautonomous forcing \begin{document}$h \in C(\mathbb{R},X)$\end{document} is \begin{document}$\mu $\end{document}-subexponentially growing for some \begin{document}$\mu >0$\end{document} (see (3.4) below for definition). Under some reasonable assumptions, we first establish an existence result for a unique nonautonomous hyperbolic equilibrium for the system in the framework of cocycle semiflows. We then demonstrate that the system exhibits a global synchronising behavior with the nonautonomous forcing \begin{document}$h$\end{document} as time varies. Finally, we apply the abstract results to stochastic partial differential equations with additive white noise and obtain stochastic hyperbolic equilibria for the corresponding systems.

A quasilinear parabolic problem with a source term and a nonlocal absorption
Hui-Ling Li, Heng-Ling Wang and Xiao-Liu Wang
2018, 17(5): 1945-1956 doi: 10.3934/cpaa.2018092 +[Abstract](291) +[HTML](144) +[PDF](313.07KB)
Abstract:

We investigate a quasi-linear parabolicproblem with nonlocal absorption, for which the comparison principle is not always available. Thesufficient conditions are established via energy method to guaranteesolution to blow up or not, and the long time behavior is alsocharacterized for global solutions.

Local Aronson-Bénilan gradient estimates and Harnack inequality for the porous medium equation along Ricci flow
Wen Wang, Dapeng Xie and Hui Zhou
2018, 17(5): 1957-1974 doi: 10.3934/cpaa.2018093 +[Abstract](362) +[HTML](160) +[PDF](333.07KB)
Abstract:

In this paper, we prove some new local Aronson-Bénilan type gradient estimates for positive solutions of the porous medium equation

coupled with Ricci flow, assuming that the Ricci curvature is bounded. As application, the related Harnack inequality is derived. Our results generalize known results. These results may be regarded as the generalizations of the gradient estimates of Lu-Ni-Vázquez-Villani and Huang-Huang-Li to the Ricci flow.

A blowup alternative result for fractional nonautonomous evolution equation of Volterra type
Pengyu Chen, Xuping Zhang and Yongxiang Li
2018, 17(5): 1975-1992 doi: 10.3934/cpaa.2018094 +[Abstract](366) +[HTML](163) +[PDF](380.25KB)
Abstract:

In this article, we consider a class of fractional non-autonomous integro-differential evolution equation of Volterra type in a Banach space \begin{document}$E$\end{document}, where the operators in linear part (possibly unbounded) depend on time \begin{document}$t$\end{document}. Combining the theory of fractional calculus, operator semigroups and measure of noncompactness with Sadovskii's fixed point theorem, we firstly proved the local existence of mild solutions for corresponding fractional non-autonomous integro-differential evolution equation. Based on the local existence result and a piecewise extended method, we obtained a blowup alternative result for fractional non-autonomous integro-differential evolution equation of Volterra type. Finally, as a sample of application, these results are applied to a time fractional non-autonomous partial integro-differential equation of Volterra type with homogeneous Dirichlet boundary condition. This paper is a continuation of Heard and Rakin [13, J. Differential Equations, 1988] and the results obtained essentially improve and extend some related conclusions in this area.

The Malgrange-Ehrenpreis theorem for nonlocal Schrödinger operators with certain potentials
Woocheol Choi and Yong-Cheol Kim
2018, 17(5): 1993-2010 doi: 10.3934/cpaa.2018095 +[Abstract](411) +[HTML](150) +[PDF](382.68KB)
Abstract:

In this paper, we prove the Malgrange-Ehrenpreis theorem for nonlocal Schrödinger operators $L_K+V$ with nonnegative potentials $V∈ L^q_{\rm{loc}}(\mathbb{R}^n)$ for $q>\frac{n}{2s}$ with $0 < s < 1$ and $n>2s$; that is to say, we obtain the existence of a fundamental solution $\mathfrak{e}_V$ for $L_K+V$ satisfying

in the distribution sense, where $\delta _0$ denotes the Dirac delta mass at the origin. In addition, we obtain a decay of the fundamental solution $\mathfrak{e}_V$.

Sharp Sobolev type embeddings on the entire Euclidean space
Angela Alberico, Andrea Cianchi, Luboš Pick and Lenka Slavíková
2018, 17(5): 2011-2037 doi: 10.3934/cpaa.2018096 +[Abstract](339) +[HTML](197) +[PDF](513.94KB)
Abstract:

A comprehensive approach to Sobolev type embeddings, involving arbitrary rearrangement-invariant norms on the entire Euclidean space \begin{document}${\mathbb R^n}$\end{document}, is offered. In particular, the optimal target space in any such embedding is exhibited. Crucial in our analysis is a new reduction principle for the relevant embeddings, showing their equivalence to a couple of considerably simpler one-dimensional inequalities. Applications to the classes of the Orlicz-Sobolev and the Lorentz-Sobolev spaces are also presented. These contributions fill in a gap in the existing literature, where sharp results in such a general setting are only available for domains of finite measure.

Exponential decay for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping
A. F. Almeida, M. M. Cavalcanti and J. P. Zanchetta
2018, 17(5): 2039-2061 doi: 10.3934/cpaa.2018097 +[Abstract](673) +[HTML](149) +[PDF](427.94KB)
Abstract:

The following coupled damped Klein-Gordon-Schrödinger equations are considered

where \begin{document}$Ω$\end{document} is a bounded domain of \begin{document}$\mathbb{R}^n$\end{document}, \begin{document}$n = 2$\end{document}, with smooth boundary \begin{document}$Γ$\end{document} and \begin{document}$ω$\end{document} is a neighbourhood of \begin{document}$\partial Ω$\end{document} satisfying the geometric control condition. Here \begin{document}$χ_{ω}$\end{document} represents the characteristic function of \begin{document}$ω$\end{document}. Assuming that \begin{document}$a, b∈ W^{1,∞}(Ω)\cap C^∞(Ω)$\end{document} are nonnegative functions such that \begin{document}$a(x) ≥ a_0 >0$\end{document} in \begin{document}$ω$\end{document} and \begin{document}$b(x) ≥ b_{0} > 0$\end{document} in \begin{document}$ω$\end{document}, the exponential decay rate is proved for every regular solution of the above system. Our result generalizes substantially the previous results given by Cavalcanti et. al in the reference [7].

On spike solutions for a singularly perturbed problem in a compact riemannian manifold
Shengbing Deng, Zied Khemiri and Fethi Mahmoudi
2018, 17(5): 2063-2084 doi: 10.3934/cpaa.2018098 +[Abstract](367) +[HTML](133) +[PDF](458.49KB)
Abstract:

Let \begin{document} $(M, g)$ \end{document} be a smooth compact riemannian manifold of dimension \begin{document} $N≥2$ \end{document} with constant scalar curvature. We are concerned with the following elliptic problem

where \begin{document} $Δ_g$ \end{document} is the Laplace-Beltrami operator on \begin{document} $M$ \end{document}, \begin{document} $p>2$ \end{document} if \begin{document} $N = 2$ \end{document} and \begin{document} $2<p<\frac{2N}{N-2}$ \end{document} if \begin{document} $N≥3$ \end{document}, \begin{document} $\varepsilon$ \end{document} is a small real parameter. We prove that there exist a function \begin{document} $Ξ$ \end{document} such that if \begin{document} $ξ_0$ \end{document} is a stable critical point of \begin{document} $Ξ(ξ)$ \end{document} there exists \begin{document} ${\varepsilon}_0>0$ \end{document} such that for any \begin{document} ${\varepsilon}∈(0,{\varepsilon}_0)$ \end{document}, problem (1) has a solution \begin{document} $u_{\varepsilon}$ \end{document} which concentrates near \begin{document} $ξ_0$ \end{document} as \begin{document} ${\varepsilon}$ \end{document} tends to zero. This result generalizes previous works which handle the case where the scalar curvature function of \begin{document} $(M,g)$ \end{document} has non-degenerate critical points.

Concentration phenomena for critical fractional Schrödinger systems
Vincenzo Ambrosio
2018, 17(5): 2085-2123 doi: 10.3934/cpaa.2018099 +[Abstract](362) +[HTML](143) +[PDF](595.68KB)
Abstract:

In this paper we study the existence, multiplicity and concentration behavior of solutions for the following critical fractional Schrödinger system

where \begin{document}$\varepsilon>0$\end{document} is a parameter, \begin{document}$s∈ (0, 1)$\end{document}, \begin{document}$N>2s$\end{document}, \begin{document}$(-Δ)^{s}$\end{document} is the fractional Laplacian operator, \begin{document}$V:\mathbb{R}^{N} \to \mathbb{R}$\end{document} and \begin{document}$W:\mathbb{R}^{N} \to \mathbb{R}$\end{document} are positive Hölder continuous potentials, \begin{document}$Q$\end{document} and \begin{document}$K$\end{document} are homogeneous \begin{document}$C^{2}$\end{document}-functions having subcritical and critical growth respectively.

We relate the number of solutions with the topology of the set where the potentials \begin{document}$V$\end{document} and \begin{document}$W$\end{document} attain their minimum values. The proofs rely on the Ljusternik-Schnirelmann theory and variational methods.

Order preservation for path-distribution dependent SDEs
Xing Huang, Chang Liu and Feng-Yu Wang
2018, 17(5): 2125-2133 doi: 10.3934/cpaa.2018100 +[Abstract](269) +[HTML](130) +[PDF](300.68KB)
Abstract:

Sufficient and necessary conditions are presented for the order preservation of path-distribution dependent SDEs. Differently from the corresponding study of distribution independent SDEs, to investigate the necessity of order preservation for the present model we need to construct a family of probability spaces in terms of the ordered pair of initial distributions.

A note concerning a property of symplectic matrices
Björn Gebhard
2018, 17(5): 2135-2137 doi: 10.3934/cpaa.2018101 +[Abstract](283) +[HTML](129) +[PDF](227.94KB)
Abstract:

This note provides a counterexample to a proposition stated in [1] regarding the neighborhood of certain \begin{document}$4× 4$\end{document} symplectic matrices.

Global behavior of bifurcation curves for the nonlinear eigenvalue problems with periodic nonlinear terms
Tetsutaro Shibata
2018, 17(5): 2139-2147 doi: 10.3934/cpaa.2018102 +[Abstract](279) +[HTML](148) +[PDF](118.91KB)
Abstract:

We consider the bifurcation problem

where \begin{document}$g(u) ∈ C^1(\mathbb{R}) $\end{document} is a periodic function with period 2π and \begin{document}$λ > 0$\end{document} is a bifurcation parameter. It is known that, under the appropriate conditions on $g$, $λ$ is parameterized by the maximum norm \begin{document}$α = \Vert u_λ\Vert_∞$\end{document} of the solution \begin{document}$u_λ$\end{document} associated with \begin{document}$λ$\end{document} and is written as \begin{document}$λ = λ(α)$\end{document}. If \begin{document}$g(u)$\end{document} is periodic, then it is natural to expect that \begin{document}$λ(α)$\end{document} is also oscillatory for \begin{document}$α \gg 1$\end{document}. We give a simple condition of \begin{document}$g(u)$\end{document}, by which we can easily check whether \begin{document}$λ(α)$\end{document} is oscillatory and intersects the line \begin{document}$λ = π^2/4$\end{document} infinitely many times for \begin{document}$\alpha \gg 1$\end{document} or not.

Double bifurcation diagrams and four positive solutions of nonlinear boundary value problems via time maps
Xuemei Zhang and Meiqiang Feng
2018, 17(5): 2149-2171 doi: 10.3934/cpaa.2018103 +[Abstract](341) +[HTML](127) +[PDF](486.17KB)
Abstract:

In this paper, we consider the existence and exactness of multiple positive solutions for the nonlinear boundary value problem

where \begin{document}$λ>0$\end{document} is a bifurcation parameter, \begin{document}$f(u)>0$\end{document} for \begin{document}$u>0$\end{document}. We give complete descriptions of the structure of bifurcation curves and determine the existence and multiplicity of positive solutions of the above problem for \begin{document}$f(u) = e^{u},\ f(u) = a^{u}(a>0),\ f(u) = u^{p}(p>0),\ f(u) = e^{u}-1,\ f(u) = a^{u}-1(a>1)$\end{document} and \begin{document}$f(u) = (1+u)^{p}(p>0)$\end{document}. Our methods are based on a detailed analysis of time maps.

Specified homogenization of a discrete traffic model leading to an effective junction condition
Nicolas Forcadel, Wilfredo Salazar and Mamdouh Zaydan
2018, 17(5): 2173-2206 doi: 10.3934/cpaa.2018104 +[Abstract](298) +[HTML](139) +[PDF](490.81KB)
Abstract:

In this paper, we focus on deriving traffic flow macroscopic models from microscopic models containing a local perturbation such as a traffic light. At the microscopic scale, we consider a first order model of the form "follow the leader" i.e. the velocity of each vehicle depends on the distance to the vehicle in front of it. We consider a local perturbation located at the origin that slows down the vehicles. At the macroscopic scale, we obtain an explicit Hamilton-Jacobi equation left and right of the origin and a junction condition at the origin (in the sense of [25]) which keeps the memory of the local perturbation. As it turns out, the macroscopic model is equivalent to a LWR model, with a flux limiting condition at the junction. Finally, we also present qualitative properties concerning the flux limiter at the junction.

Local inversion of a class of piecewise regular maps
Laura Poggiolini and Marco Spadini
2018, 17(5): 2207-2224 doi: 10.3934/cpaa.2018105 +[Abstract](314) +[HTML](122) +[PDF](1165.64KB)
Abstract:

This paper provides sufficient conditions for any map L, that is strongly piecewise linear relatively to a decomposition of \begin{document}$\mathbb{R}^k$\end{document} in admissible cones, to be invertible. Namely, via a degree theory argument, we show that when there are at most four convex pieces (or three pieces with at most a non convex one), the map is invertible. Examples show that the result cannot be plainly extended to a greater number of pieces. Our result is obtained by studying the structure of strongly piecewise linear maps. We then extend the results to the PC1 case.

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