# American Institute of Mathematical Sciences

ISSN:
1534-0392

eISSN:
1553-5258

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## Communications on Pure & Applied Analysis

July 2017 , Volume 16 , Issue 4

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Xiaojie Hou and
2017, 16(4): 1103-1120 doi: 10.3934/cpaa.2017053 +[Abstract](1197) +[HTML](11) +[PDF](410.88KB)
Abstract:

This paper studies the traveling wave solutions to a three species competition cooperation system, which is derived from a spatially averaged and temporally delayed Lotka Volterra system. The existence of the traveling waves is investigated via a new type of monotone iteration method. The upper and lower solutions come from either the waves of KPP equation or those of certain two species Lotka Volterra system. We also derive the asymptotics and uniqueness of the wave solutions.

2017, 16(4): 1121-1134 doi: 10.3934/cpaa.2017054 +[Abstract](1314) +[HTML](7) +[PDF](356.3KB)
Abstract:

The purpose of this paper is to investigate positive solutions of integral equations involving Bessel potential. Exploiting the moving plane method in integral form, we give the radial symmetry of both the domain and solutions of our integral equations in exterior domains and in annular domains respectively.

2017, 16(4): 1135-1146 doi: 10.3934/cpaa.2017055 +[Abstract](962) +[HTML](8) +[PDF](396.52KB)
Abstract:

As shown by Dancer's counterexample [12], one cannot expect to have (generically) more than five solutions to the semilinear boundary value problem

when $f(0, x)=0$ and $\partial f/\partial u$ crosses the first two eigenvalues of $-\Delta$ with Dirichlet boundary conditions on $\partial B$, as $|u|$ grows from $0$ to $\infty$. Despite the fact that the nonlinearity $f$ in [12] can be taken "arbitrarily close" to autonomous, we prove that the dependence of $f$ on $x$ is indeed essential in the arguments. More precisely, we show that (P) with $f$ independent of $x$ and satisfying the assumptions in [12] has at least six, and generically seven solutions. Under more stringent conditions on the non-linearity, we prove that there are up to nine solutions. Importantly, we do not assume any symmetry on $f$ for any of our results.

2017, 16(4): 1147-1168 doi: 10.3934/cpaa.2017056 +[Abstract](1127) +[HTML](6) +[PDF](477.74KB)
Abstract:

We study a nonlinear Dirichlet problem driven by the sum of a $p-$Laplacian ($p>2$) and a Laplacian and which at $\pm\infty$ is resonant with respect to the spectrum of $\left(-\triangle_{p}, W_{0}^{1, p}\left(\Omega\right)\right)$ and at zero is resonant with respect to the spectrum of $\left(-\triangle, H_{0}^{1}\left(\Omega\right) \right)$ (double resonance). We prove two multiplicity theorems providing three and four nontrivial solutions respectivelly, all with sign information. Our approach uses critical point theory together with truncation and comparison techniques and Morse theory.

2017, 16(4): 1169-1188 doi: 10.3934/cpaa.2017057 +[Abstract](1421) +[HTML](4) +[PDF](472.01KB)
Abstract:

We study the existence of positive solutions for perturbations of the classical eigenvalue problem for the Dirichlet $p-$Laplacian. We consider three cases. In the first the perturbation is $(p-1)-$sublinear near $+\infty$, while in the second the perturbation is $(p-1)-$superlinear near $+\infty$ and in the third we do not require asymptotic condition at $+\infty$. Using variational methods together with truncation and comparison techniques, we show that for $\lambda\in (0, \widehat{\lambda}_1)$ -$\lambda>0$ is the parameter and $\widehat{\lambda}_1$ being the principal eigenvalue of $\left(-\Delta_p, W^{1, p}_0(\Omega)\right)$ -we have positive solutions, while for $\lambda\geq \widehat{\lambda}_1$, no positive solutions exist. In the "sublinear case" the positive solution is unique under a suitable monotonicity condition, while in the "superlinear case" we produce the existence of a smallest positive solution. Finally, we point out an existence result of a positive solution without requiring asymptotic condition at $+\infty$, provided that the perturbation is damped by a parameter.

2017, 16(4): 1189-1198 doi: 10.3934/cpaa.2017058 +[Abstract](1144) +[HTML](10) +[PDF](370.14KB)
Abstract:

We devote the present paper to studying a critical exponent with respect to the stability of solutions to Hénon type equation on the hyperbolic space. In order to specify the critical exponent, we prove the existence and nonexistence result for stable solutions. In this paper, we obtain stable, positive, and radial solutions of the Hénon type equation for the supercritical case. Moreover, we prove that the set of these stable solutions has the separation structure.

2017, 16(4): 1199-1232 doi: 10.3934/cpaa.2017059 +[Abstract](1135) +[HTML](6) +[PDF](553.45KB)
Abstract:

We discuss the existence of a Fredholm–type Alternative for a recurrent second order linear equation, which is disconjugate in a strong sense. The basic result is about bounded solutions of equations with bounded coefficients: it depends on kinematic similarities that allow to reduce the problem to a pair of very simple normal forms. Then the result is specialized to recurrent equations, by means of Favard theory.

2017, 16(4): 1233-1252 doi: 10.3934/cpaa.2017060 +[Abstract](1156) +[HTML](3) +[PDF](446.7KB)
Abstract:

We consider a nonlinear Schrödinger equation with a delta-function impurity at the origin of the space domain. We study the asymptotic behavior of the solutions with the theory of infinite dynamical system. We first prove the existence of a global attractor in $H^1_0(-1, 1)$. We also establish that this global attractor is a compact subset of $H^{\frac{3}{2}-\epsilon}(-1, 1)$.

2017, 16(4): 1253-1264 doi: 10.3934/cpaa.2017061 +[Abstract](1461) +[HTML](6) +[PDF](416.83KB)
Abstract:

In this paper, we investigate the Hardy-Sobolev type integral systems (1) with Dirichlet boundary conditions in a half space $\mathbb{R}_+^n$. We use the method of moving planes in integral forms introduced by Chen, Li and Ou [12] to prove that each pair of integrable positive solutions $(u, v)$ of the above integral system is rotationally symmetric about $x_n$-axis in both subcritical and critical cases $\frac{n-t}{p+1}+\frac{n-s}{q+1}\geq n-2m$ (Theorem 1.1). We also derive the non-existence of nontrivial nonnegative solutions with finite energy in the subcritical case (Theorem 1.2). By slightly modifying our arguments for studying the integral system, we can prove by a similar but simpler way that the same conclusions also hold for a single integral equation of Hardy-Sobolev type in both critical and subcritical cases (Theorem 1.3).

2017, 16(4): 1265-1292 doi: 10.3934/cpaa.2017062 +[Abstract](1167) +[HTML](7) +[PDF](507.69KB)
Abstract:

In this paper, we study the asymptotic behavior of solutions to the initial boundary value problem for the micropolar fluid model in half line $\mathbb{R}_{+}:=(0, \infty).$ Inspired by the relationship between the micropolar fluid model and Navier-Stokes system, we can prove that the composite wave consisting of the subsonic BL-solution, the contact wave, and the rarefaction wave for the inflow problem on micropolar fluid model is time-asymptotically stable. Meanwhile, we obtain the global existence of solutions based on the basic energy method.

2017, 16(4): 1293-1314 doi: 10.3934/cpaa.2017063 +[Abstract](999) +[HTML](4) +[PDF](466.96KB)
Abstract:

We consider a parametric semilinear Robin problem driven by the Laplacian plus an indefinite potential. The reaction term involves competing nonlinearities. More precisely, it is the sum of a parametric sublinear (concave) term and a superlinear (convex) term. The superlinearity is not expressed via the Ambrosetti-Rabinowitz condition. Instead, a more general hypothesis is used. We prove a bifurcation-type theorem describing the set of positive solutions as the parameter $\lambda > 0$ varies. We also show the existence of a minimal positive solution $\tilde{u}_\lambda$ and determine the monotonicity and continuity properties of the map $\lambda \mapsto \tilde{u}_\lambda$.

2017, 16(4): 1315-1330 doi: 10.3934/cpaa.2017064 +[Abstract](1100) +[HTML](5) +[PDF](464.19KB)
Abstract:

We consider a quasi-linear elliptic equation with Dirac source terms arising in a generalized self-dual Chern-Simons-Higgs gauge theory. In this paper, we study doubly periodic vortices with arbitrary vortex configuration. First of all, we show that under doubly periodic condition, there are only two types of solutions, topological and non-topological solutions as the coupling parameter goes to zero. Moreover, we succeed to construct non-topological solution with $k$ bubbles where $k\in\mathbb{N}$ is any given number. To find a solution, we analyze the structure of quasi-linear elliptic equation carefully and apply the method developed in the recent work [16].

2017, 16(4): 1331-1372 doi: 10.3934/cpaa.2017065 +[Abstract](1144) +[HTML](5) +[PDF](655.15KB)
Abstract:

A three-dimensional thermo-visco-elastic system for Kelvin-Voigt type material at small strains is considered. The system involves nonlinear temperature-dependent specific heat relevant in the limit of low temperature range. The existence of a unique global regular solution is proved without small data assumptions. The proof consists of two parts. First the existence of a local in time solution is proved by the Banach successive approximations method. Then a lower bound on temperature and global a priori estimates are derived with the help of the theory of anisotropic Sobolev spaces with a mixed norm. Such estimates allow to extend the local solution step by step in time. The paper generalizes the results of the previous author's publication in SIAM J. Math. Anal. 45, No. 4 (2013), pp. 1997–2045.

2017, 16(4): 1373-1404 doi: 10.3934/cpaa.2017066 +[Abstract](1445) +[HTML](11) +[PDF](1600.21KB)
Abstract:

This paper deals with a competition model with dynamically allocated toxin production in the unstirred chemostat. First, the existence and uniqueness of positive steady state solutions of the single population model is attained by the general maximum principle, spectral analysis and degree theory. Second, the existence of positive equilibria of the two-species system is investigated by the degree theory, and the structure and stability of nonnegative equilibria of the two-species system are established by the bifurcation theory. The results show that stable coexistence solution can occur with dynamic toxin production, which cannot occur with constant toxin production. Biologically speaking, it implies that dynamically allocated toxin production is sufficiently effective in the occurrence of coexisting. Finally, numerical results illustrate that a wide variety of dynamical behaviors can be achieved for the system with dynamic toxin production, including competition exclusion, bistable attractors, stable positive equilibria and stable limit cycles, which complement the analytic results.

2017, 16(4): 1405-1426 doi: 10.3934/cpaa.2017067 +[Abstract](1339) +[HTML](3) +[PDF](457.14KB)
Abstract:

A diffusive competing pioneer and climax system without cooperative property is investigated. We consider a special case in which the system has no co-existence equilibrium. Under the appropriate assumptions, we show the linear determinacy and the existence of single spreading speed. Furthermore, we obtain the existence of traveling wave solution which connects two boundary equilibria, and also confirm that the spreading speed coincides with the minimal wave speed. The results in this article reveals a phenomenon of strongly biological invasion which implies that the invasion of a new species will leads to the extinction of the resident species.

2017, 16(4): 1427-1454 doi: 10.3934/cpaa.2017068 +[Abstract](1072) +[HTML](4) +[PDF](948.77KB)
Abstract:

We study various regularity properties of minimizers of the $\Phi$–perimeter, where $\Phi$ is a norm. Under suitable assumptions on $\Phi$ and on the dimension of the ambient space, we prove that the boundary of a cartesian minimizer is locally a Lipschitz graph out of a closed singular set of small Hausdorff dimension. Moreover, we show the following anisotropic Bernstein-type result: any entire cartesian minimizer is the subgraph of a monotone function depending only on one variable.

2017, 16(4): 1455-1470 doi: 10.3934/cpaa.2017069 +[Abstract](1273) +[HTML](2) +[PDF](598.92KB)
Abstract:

We study the behavior of the solution of a generalized damped KdV equation $u_t + u_x + u_{xxx} + u^p u_x + \mathscr{L}_{\gamma}(u)= 0$. We first state results on the local well-posedness. Then when $p \geq 4$, conditions on $\mathscr{L}_{\gamma}$ are given to prevent the blow-up of the solution. Finally, we numerically build such sequences of damping.

2017, 16(4): 1471-1492 doi: 10.3934/cpaa.2017070 +[Abstract](965) +[HTML](2) +[PDF](665.03KB)
Abstract:

We study the set of $T$-periodic solutions of a class of $T$-periodically perturbed coupled and nonautonomous differential equations on manifolds. By using degree-theoretic methods we obtain a global continuation result for the $T$-periodic solutions of the considered equations.

2017, 16(4): 1493-1516 doi: 10.3934/cpaa.2017071 +[Abstract](976) +[HTML](2) +[PDF](457.53KB)
Abstract:

We consider the physical problem of protecting a thermally conducting body from overheating by thermal barrier coatings on a bounded domain, which has two components with a thin coating surrounding the body (of metallic nature), subject to the Dirichlet boundary condition. The coating is composed of two layers, the pure ceramic part and the mixed part. The latter is assumed to be functionally graded material (FGM) that is meant to make a smooth transition from being metallic to being ceramic. The thermal tensor $A$ is isotropic on the body, and anisotropic on the coating; and the size of thermal tensor may differ significantly in these components. Eigenfunction expansion of the interior temperature function indicates that small eigenvalues of the elliptic operator $u\mapsto -\nabla\cdot \left(A\nabla u\right)$ are desirable for the insulation of the body. Therefore, we are motivated to study the asymptotic behavior of the eigenpairs of the Dirichelt eigenvalue problem, as the thickness of the coating shrinks. Our results greatly generalize those by Rosencrans and Wang [8] where the case of single coating layer is considered. In particular, we find new optimal scaling relationship between the thickness of the coating and its thermal conductivity that guarantees at least the principal eigenvalue is small for any general FGMs.

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