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1078-0947

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1553-5231

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## Discrete & Continuous Dynamical Systems - A

February 2016 , Volume 36 , Issue 2

Special issue dedicated to the Late Professor Rou-Huai Wang on the occasion of his 90th birthday

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2016, 36(2): i-iii
doi: 10.3934/dcds.2016.36.2i

*+*[Abstract](1869)*+*[PDF](1914.3KB)**Abstract:**

Professor Rou-Huai Wang (October 30, 1924 - November 5, 2001) was a mathematician who proved fundamental results for partial differential equations, helped to introduce modern PDE theory to the Chinese mathematics community since early 50's, and played a leading role in revitalizing PDE research in China after the disastrous "Cultural Revolution" (1966-1976). He was regarded by many Chinese mathematicians of younger generations as a visionary, generous and caring mentor.

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2016, 36(2): 577-600
doi: 10.3934/dcds.2016.36.577

*+*[Abstract](2262)*+*[PDF](488.2KB)**Abstract:**

In this paper, we apply the sharp Adams-type inequalities for the Sobolev space $W^{m,\frac{n}{m}}\left( \mathbb{R} ^{n}\right) $ for any positive real number $m$ less than $n$, established by Ruf and Sani [46] and Lam and Lu [30,31], to study polyharmonic equations in $\mathbb{R}^{2m}$. We will consider the polyharmonic equations in $\mathbb{R}^{2m}$ of the form \[ \left( I-\Delta\right) ^{m}u=f(x,u)\text{ in }% \mathbb{R} ^{2m}. \] We study the existence of the nontrivial solutions when the nonlinear terms have the critical exponential growth in the sense of Adams' inequalities on the entire Euclidean space. Our approach is variational methods such as the Mountain Pass Theorem ([5]) without Palais-Smale condition combining with a version of a result due to Lions ([39,40]) for the critical growth case. Moreover, using the regularity lifting by contracting operators and regularity lifting by combinations of contracting and shrinking operators developed in [14] and [11], we will prove that our solutions are uniformly bounded and Lipschitz continuous. Finally, using the moving plane method of Gidas, Ni and Nirenberg [22,23] in integral form developed by Chen, Li and Ou [12] together with the Hardy-Littlewood-Sobolev type inequality instead of the maximum principle, we prove our positive solutions are radially symmetric and monotone decreasing about some point. This appears to be the first work concerning existence of nontrivial nonnegative solutions of the Bessel type polyharmonic equation with exponential growth of the nonlinearity in the whole Euclidean space.

2016, 36(2): 601-609
doi: 10.3934/dcds.2016.36.601

*+*[Abstract](2746)*+*[PDF](363.3KB)**Abstract:**

We consider positive semistable solutions $u$ of $Lu+f(u)=0$ with zero Dirichlet boundary condition, where $L$ is a uniformly elliptic operator and $f\in C^2$ is a positive, nondecreasing, and convex nonlinearity which is superlinear at infinity. Under these assumptions, the boundedness of all semistable solutions is expected up to dimension $n\leq 9$, but only established for $n\leq 4$.

In this paper we prove the $L^\infty$ bound up to dimension $n=5$ under the following further assumption on $f$: for every $\varepsilon>0$, there exist $T=T(\varepsilon)$ and $C=C(\varepsilon)$ such that $f'(t)\leq Cf(t)^{1+\varepsilon}$ for all $t>T$. This bound will follow from a $L^p$-estimate for $f'(u)$ for every $p<3$ (and for all $n\geq 2$). Under a similar but more restrictive assumption on $f$, we also prove the $L^\infty$ estimate when $n=6$. We remark that our results do not assume any lower bound on $f'$.

2016, 36(2): 611-629
doi: 10.3934/dcds.2016.36.611

*+*[Abstract](2056)*+*[PDF](447.0KB)**Abstract:**

In this paper, the compressible Navier-Stokes-Korteweg system with friction is considered in $\mathbb{R}^3$. Via the linear analysis, we show the existence, uniqueness and time-asymptotic stability of the time periodic solution when a time periodic external force is taken into account. Our proof is based on a combination of the energy method and the contraction mapping theorem. In particular, this is the first paper that a time periodic solution can be obtained in the whole space $\mathbb{R}^3$ only under the suitable smallness condition of $H^{N-1}\cap L^1$--norm$(N\geq5)$ of time periodic external force.

2016, 36(2): 631-642
doi: 10.3934/dcds.2016.36.631

*+*[Abstract](2494)*+*[PDF](408.3KB)**Abstract:**

This paper is concerned with large time behavior of solutions for the Cauchy problem of a semilinear pseudo-parabolic equation with small perturbation. It is revealed that small perturbation may develop large variation of solutions with the evolution of time, which is similar to that for the standard heat equation with nonlinear sources.

2016, 36(2): 643-652
doi: 10.3934/dcds.2016.36.643

*+*[Abstract](2448)*+*[PDF](362.8KB)**Abstract:**

This paper is devoted to the following second order dissipative dynamical system \begin{equation*} u''+cu'+ \nabla g(u)+h(u)=e(t) ~\mbox{in}~\mathbb{R}^n. \end{equation*} When $g(u)=g(|u|)$, $\nabla g$ is a coercive function and $h$ is bounded, we use the coincidence degree theory to obtain some existence results of rotating periodic solutions, i.e., $u(t+T)=Qu(t)$, $\forall t\in \mathbb{R}$, with $T>0$ and $Q$ an orthogonal matrix, for $g$ to be nonsingular and singular at zero respectively. Specially, when some strong force type assumption is supposed on $g$, we obtain some new existence results of non-collision solutions for singular systems.

2016, 36(2): 653-660
doi: 10.3934/dcds.2016.36.653

*+*[Abstract](1968)*+*[PDF](329.8KB)**Abstract:**

In this paper, we study the existence of smooth local solutions to Weingarten equations and $\sigma_k$-equations. We will prove that, for $2\le k\le n-1$, the Weingarten equations and the $\sigma_k$-equations always have smooth local solutions regardless of the sign of the functions in the right-hand side of the equations. We will demonstrate that the associated linearized equations are uniformly elliptic if we choose the initial approximate solutions appropriately.

2016, 36(2): 661-682
doi: 10.3934/dcds.2016.36.661

*+*[Abstract](2631)*+*[PDF](477.7KB)**Abstract:**

This article studies the initial boundary value problem for a class of semilinear edge-degenerate parabolic equations with singular potential term. By introducing a family of potential wells, we derive a threshold of the existence of global solutions with exponential decay, and the blow-up in finite time in both cases with low initial energy and critical initial energy.

2016, 36(2): 683-699
doi: 10.3934/dcds.2016.36.683

*+*[Abstract](2880)*+*[PDF](434.6KB)**Abstract:**

In this paper, we study the existence of positive solution for the following p-Laplacain type equations with critical nonlinearity \begin{equation*} \left\{ \renewcommand{\arraystretch}{1.25} \begin{array}{ll} -\Delta_p u + V （x)|u|^{p-2}u = K(x)f(u)+P(x)|u|^{p^*-2}u, \quad x\in\mathbb{R}^N,\\ u \in \mathcal{D}^{1,p}(\mathbb{R}^N), \end{array} \right. \end{equation*} where $\Delta_p u = div(|\nabla u|^{p-2} \nabla u),\ 1 < p < N,\ p^* =\frac {Np}{N-p}$, $V(x)$, $K(x)$ are positive continuous functions which vanish at infinity, $f$ is a function with a subcritical growth, and $P(x)$ is a bounded, nonnegative continuous function. By working in the weighted Sobolev spaces, and using variational method, we prove that the given problem has at least one positive solution.

2016, 36(2): 701-714
doi: 10.3934/dcds.2016.36.701

*+*[Abstract](2514)*+*[PDF](392.1KB)**Abstract:**

We apply some new ideas to derive $C^2$ estimates for solutions of a general class of fully nonlinear elliptic equations on Riemannian manifolds under a ``minimal'' set of assumptions which are standard in the literature. Based on these estimates we solve the Dirichlet problem using the continuity method and degree theory.

2016, 36(2): 715-730
doi: 10.3934/dcds.2016.36.715

*+*[Abstract](2619)*+*[PDF](472.8KB)**Abstract:**

The authors of this paper study some singular phenomena (blowing-up or vanishing in finite time) of solutions to the initial homogeneous $\hbox{Neumann}$ problem of a nonlinear diffusion equation involving the $p(x,t)$-Laplace operator and a nonlinear source. The variable exponent $p(x,t)$ leads to the failure of some techniques, such as upper-lower solutions technique and the scaling method etc., in studying the problem; it also leads to the lack of some valuable properties such as the monotonicity of the energy integral etc. The authors construct a suitable control functional, improve the regularity of the approximate solutions and obtain a new energy inequality to prove that the solution of the problem with a positive initial energy blows up in finite time. Furthermore, under some appropriate conditions, the authors study the vanishing property and the extinction rate estimate of the solutions to the problem by establishing some inequalities the solutions satisfy. It is worth pointing out that the results are obtained with the assumption that $p_{t}(x,t)$ is only negative and integrable which is weaker than those the most of the other papers required.

2016, 36(2): 731-762
doi: 10.3934/dcds.2016.36.731

*+*[Abstract](2492)*+*[PDF](627.7KB)**Abstract:**

We study the existence, concentration and multiplicity of weak solutions to the quasilinear Schrödinger equation with critical Sobolev growth \begin{equation*} \left\{ \begin{gathered} - {\varepsilon ^2}\Delta u + V(x)u - {\varepsilon ^2}\Delta (u^2)u = W(x){u^{q - 1}} + {u^{2\cdot{2^*} - 1}} {\text{ in }}{\mathbb{R}^N},\\ u > 0{\text{ in }}{\mathbb{R}^N},\\ \end{gathered} \right. \end{equation*} where $\varepsilon $ is a small positive parameter, $N \ge 3$, ${2^ * } = \frac{{2N}} {{N - 2}}$, $4 < q < 2 \cdot {2^ * }$, $\min V > 0$ and $\inf W > 0$. Under proper assumptions, we obtain the existence and concentration phenomena of soliton solutions of the above problem. With minimax theorems and Ljusternik-Schnirelmann theory, we also obtain multiple soliton solutions by employing the topology of the set where the potentials $V(x)$ attains its minimum and $W(x)$ attains its maximum.

2016, 36(2): 763-784
doi: 10.3934/dcds.2016.36.763

*+*[Abstract](2124)*+*[PDF](476.0KB)**Abstract:**

The present paper is devoted to studying the Hill-type formula and Krein-type trace formula for ODE, which is a continuous work of our previous work for Hamiltonian systems [5]. Hill-type formula and Krein-type trace formula are given by Hill at 1877 and Krein in 1950's separately. Recently, we find that there is a closed relationship between them [5]. In this paper, we will obtain the Hill-type formula for the $S$-periodic orbits of the first order ODEs. Such a kind of orbits is considered naturally to study the symmetric periodic and quasi-periodic solutions. By some similar idea in [5], based on the Hill-type formula, we will build up the Krein-type trace formula for the first order ODEs, which can be seen as a non-self-adjoint version of the case of Hamiltonian system.

2016, 36(2): 785-803
doi: 10.3934/dcds.2016.36.785

*+*[Abstract](2446)*+*[PDF](429.0KB)**Abstract:**

The existence of $2\pi$-periodic positive solutions of the equation $$ u'' + u = \displaystyle{\frac{a(x)}{u^3}} $$ is studied, where $a$ is a positive smooth $2\pi$-periodic function. Under some non-degenerate conditions on $a$, the existence of $2\pi$-periodic solutions to the equation is established.

2016, 36(2): 805-832
doi: 10.3934/dcds.2016.36.805

*+*[Abstract](2502)*+*[PDF](538.0KB)**Abstract:**

Existence, uniqueness, and stability of Heaviside function like solutions of a Keller and Segel's minimal chemotaxis model are established when a chemotaxis parameter is large enough. Asymptotic expansions of the solution in terms of the large chemotaxis parameter are also derived.

2016, 36(2): 833-849
doi: 10.3934/dcds.2016.36.833

*+*[Abstract](2062)*+*[PDF](421.1KB)**Abstract:**

We study the asymptotic relation of solutions between the hyperbolic equation and the parabolic one over a one-dimensional bounded interval, both of which model a simple electrostatic micro-electro-mechanical system (MEMS) device. The relation is characterized by a limit as a physical parameter representing the strength of inertial forces tends to zero. We call this limit the viscosity dominated limit. It is shown that in this singular limit the solution of the hyperbolic model converges to that of the parabolic one globally in time. Also the higher order terms including the initial layer corrections, as well as the related error estimates, are derived. Furthermore, it is proved that the convergence is valid for global solutions with large initial data.

2016, 36(2): 851-860
doi: 10.3934/dcds.2016.36.851

*+*[Abstract](2243)*+*[PDF](367.5KB)**Abstract:**

We are concerned with the existence of positive solutions for a coupled Schrödinger system \begin{equation*} \left\{ \begin{aligned} &-\Delta{u}_1+{\lambda}_1 {u}_1={\mu}_1 {u}_1^3+\varepsilon \beta(x) {u}_1 {u}_2^2 & ~~in &~~~~ \mathbb{R}^3,\\ &-\Delta{u}_2+{\lambda}_2 {u}_2={\mu}_2 {u}_2^3+\varepsilon \beta(x) {u}_1^2 {u}_2 & ~~in & ~~~~\mathbb{R}^3,\\ &{u}_1>0, ~~{u}_2>0& ~~in & ~~~~\mathbb{R}^3,\\ &{u}_1\in H^1(\mathbb{R}^3),~~{u}_2\in H^1(\mathbb{R}^3), \end{aligned} \right. \end{equation*} where ${\lambda}_1,{\lambda}_2,{\mu}_1,{\mu}_2$ are positive constants. We use perturbation methods to prove that if $\beta \in L^r(\mathbb{R}^3)(r\geq 3)$ doesn't change sign, as corresponding $\varepsilon$ is sufficiently small the system has a positive solution of which both components are positive. Our results is also true for domain $\mathbb{R}^{2}$ and for domain $ \mathbb {R}^{N}, N \geq 4 $ when the similar system is subcritical.

2016, 36(2): 861-875
doi: 10.3934/dcds.2016.36.861

*+*[Abstract](2445)*+*[PDF](412.5KB)**Abstract:**

We prove the global-in-time existence of intermediate weak solutions of the equations of chemotaxis system in a bounded domain of $\mathbb{R}^2$ or $\mathbb{R}^3$ with initial chemical concentration small in $H^1$. No smallness assumption is imposed on the initial cell density which is in $L^2$. We first show that when the initial chemical concentration $c_0$ is small only in $H^1$ and $(n_0-n_\infty,c_0)$ is smooth, the classical solution exists for all time. Then we construct weak solutions as limits of smooth solutions corresponding to mollified initial data. Finally we determine the asymptotic behavior of the global solutions.

2016, 36(2): 877-893
doi: 10.3934/dcds.2016.36.877

*+*[Abstract](1804)*+*[PDF](430.6KB)**Abstract:**

In this paper, let $n\geq2$ be an integer, $P=diag(-I_{n-\kappa},I_\kappa,-I_{n-\kappa}, I_\kappa)$ for some integer $\kappa\in[0, n-1)$, and $\Sigma \subset {\bf R}^{2n}$ be a partially symmetric compact convex hypersurface, i.e., $x\in \Sigma$ implies $Px\in\Sigma$. We prove that if $\Sigma$ is $(r,R)$-pinched with $\frac{R}{r}<\sqrt{\frac{5}{3}}$, then $\Sigma$ carries at least two geometrically distinct P-symmetric closed characteristics which possess at least $2n-4\kappa$ Floquet multipliers on the unit circle of the complex plane.

2016, 36(2): 895-916
doi: 10.3934/dcds.2016.36.895

*+*[Abstract](1911)*+*[PDF](470.3KB)**Abstract:**

In this paper we prove the existence of classical solutions to near field reflector problems, both for a point light source and for a parallel light source, with planar receivers. These problems involve Monge-Ampère type equations, subject to nonlinear oblique boundary conditions. Our approach builds on earlier work in the optimal transportation case by Trudinger and Wang and makes use of a recent extension of degree theory to oblique boundary conditions by Li, Liu and Nguyen.

Infinitely many positive and sign-changing solutions for nonlinear fractional scalar field equations

2016, 36(2): 917-939
doi: 10.3934/dcds.2016.36.917

*+*[Abstract](3247)*+*[PDF](519.4KB)**Abstract:**

We consider the following nonlinear fractional scalar field equation $$ (-\Delta)^s u + u = K(|x|)u^p,\ \ u > 0 \ \ \hbox{in}\ \ \mathbb{R}^N, $$ where $K(|x|)$ is a positive radial function, $N\ge 2$, $0 < s < 1$, and $1 < p < \frac{N+2s}{N-2s}$. Under various asymptotic assumptions on $K(x)$ at infinity, we show that this problem has infinitely many non-radial positive solutions and sign-changing solutions, whose energy can be made arbitrarily large.

2016, 36(2): 941-952
doi: 10.3934/dcds.2016.36.941

*+*[Abstract](1762)*+*[PDF](360.5KB)**Abstract:**

In [10], using a Theorem of Clark and in [1] several multiplicity results were obtained for families of semilinear elliptic partial differential equations. Here these results are extended so as to include more general spatially heterogeneous models arising in population dynamics. The optimality of the general assumptions imposed to get some of these multiplicity results is also analyzed.

2016, 36(2): 953-969
doi: 10.3934/dcds.2016.36.953

*+*[Abstract](3639)*+*[PDF](400.1KB)**Abstract:**

We study a two-species Lotka-Volterra competition model in an advective homogeneous environment. It is assumed that two species have the same population dynamics and diffusion rates but different advection rates. We show that if one competitor disperses by random diffusion only and the other assumes both random and directed movements, then the one without advection prevails. If two competitors are drifting along the same direction but with different advection rates, then the one with the smaller advection rate wins. Finally we prove that if the two competitors are drifting along the opposite direction, then two species will coexist. These results imply that the movement without advection in homogeneous environment is evolutionarily stable, as advection tends to move more individuals to the boundary of the habitat and thus cause the distribution of species mismatch with the resources which are evenly distributed in space.

2016, 36(2): 971-980
doi: 10.3934/dcds.2016.36.971

*+*[Abstract](2338)*+*[PDF](369.3KB)**Abstract:**

Consider the existence of rotationally symmetric solutions to the $L_p$-Minkowski problem for $p=-n-1$. Recently a sufficient condition was obtained for the existence via the variational method and a blow-up analysis in [16]. In this paper we use a topological degree method to prove the same existence and show the result holds under a similar complementary sufficient condition. Moreover, by this degree method, we obtain the existence result in a perturbation case.

2016, 36(2): 981-1004
doi: 10.3934/dcds.2016.36.981

*+*[Abstract](2366)*+*[PDF](489.1KB)**Abstract:**

Under the conditions of both the initial data being the small perturbation of given steady state solution and the boundary strength being small, the global existence of smooth solution to the initial boundary value problem of the relativistic Euler-Poisson equations is proved. The convergence of the global smooth solution to smooth steady state solution in time exponentially is also obtained when time goes to infinity.

2016, 36(2): 1005-1021
doi: 10.3934/dcds.2016.36.1005

*+*[Abstract](2153)*+*[PDF](420.7KB)**Abstract:**

In this paper, the existence and stability results for a two-parameter family of vector solitary-wave solutions (

*i.e*both components are nonzero) of the nonlinear Schrödinger system \begin{equation*} \left\{ \begin{matrix} iu_t+ u_{xx} + (a |u|^2 + b |v|^2) u=0,\\ iv_t+ v_{xx} + (b |u|^2 + c |v|^2) v=0,\\ \end{matrix} \right. \end{equation*} where $u,v$ are complex-valued functions of $(x,t)\in \mathbb R^2$, and $a,b,c \in \mathbb R$ are established. The results extend our earlier ones as well as those of Ohta, Cipolatti and Zumpichiatti and de Figueiredo and Lopes. As opposed to other methods used before to establish existence and stability where the two constraints of the minimization problems are related to each other, our approach here characterizes solitary-wave solutions as minimizers of an energy functional subject to two independent constraints. The set of minimizers is shown to be stable; and depending on the interplay between the parameters $a,b$ and $c$, further information about the structures of this set are given.

2016, 36(2): 1023-1039
doi: 10.3934/dcds.2016.36.1023

*+*[Abstract](1881)*+*[PDF](390.5KB)**Abstract:**

In this work, we study the existence of $C^{\infty}$ local solutions to $2$-Hessian equation in $\mathbb{R}^{3}$. We consider the case that the right hand side function $f$ possibly vanishes, changes the sign, is positively or negatively defined. We also give the convexities of solutions which are related with the annulation or the sign of right-hand side function $f$. The associated linearized operator are uniformly elliptic.

2016, 36(2): 1041-1060
doi: 10.3934/dcds.2016.36.1041

*+*[Abstract](2434)*+*[PDF](423.5KB)**Abstract:**

This paper concerns the boundary behavior and the asymptotic behavior of solutions to a class of boundary-initial parabolic problems with boundary degeneracy. At the degenerate boundary, it is shown that the diffusion vanishes and the solution possesses the invariability if the degeneracy is sufficiently strong. As to the asymptotic behavior, it is proved that the decay rate is an exponential function if the degeneracy is weak enough, while a power function if it is not.

2016, 36(2): 1061-1084
doi: 10.3934/dcds.2016.36.1061

*+*[Abstract](2780)*+*[PDF](508.6KB)**Abstract:**

This note is devoted to the discussion on the existence and blow up of the solutions to the parabolic elliptic type Patlak-Keller-Segel system on the whole space case. The problem in two dimension is closely related to the Logarithmic Hardy-Littlewood-Sobolev inequality, which directly introduced the critical mass $8\pi$. While in the higher dimension case, it is related to the Hardy-Littlewood-Sobolev inequality. Therefore, a porous media type nonlinear diffusion has been introduced in order to balance the aggregation. We will review the critical exponents which were introduced in the literature, namely, the exponent $m=2-2/n$ which comes from the scaling invariance of the mass, and the exponent $m=2n/(n+2)$ which comes from the conformal invariance of the entropy. Finally a new result on the model with a general potential, inspired from the Hardy-Littlewood-Sobolev inequality, will be given.

2016, 36(2): 1085-1103
doi: 10.3934/dcds.2016.36.1085

*+*[Abstract](3126)*+*[PDF](441.4KB)**Abstract:**

In this paper we present a sufficient condition for the global well-posedness of classical solutions to an initial value problem of compressible isentropic Navier-Stokes equations in the whole space $\mathbb{R}^3$. As an immediate result, the main theorem obtained implies that the Cauchy problem of compressible Navier-Stokes equations with vacuum has a global unique classical solution, provided the initial energy is sufficiently small, or the shear viscosity coefficient is sufficiently large, or the upper bound of the initial density is suitably small and the adiabatic exponent $\gamma\in (1,3/2)$. These results particularly extend the recent ones due to Huang-Li-Xin [7], where the global well-posedness of classical solutions with small initial energy was established.

2016, 36(2): 1105-1124
doi: 10.3934/dcds.2016.36.1105

*+*[Abstract](2042)*+*[PDF](474.0KB)**Abstract:**

Based on the theory of the local exact boundary controllability for first order quasilinear hyperbolic systems, using an extension method, the authors establish the exact controllability in a shorter time by means of internal controls acting on suitable domains. In particular, under certain special but reasonable hypotheses, the exact controllability can be realized only by internal controls, and the control time can be arbitrarily small.

2016, 36(2): 1125-1141
doi: 10.3934/dcds.2016.36.1125

*+*[Abstract](4172)*+*[PDF](441.9KB)**Abstract:**

In this paper, we consider the following system of pseudo-differential nonlinear equations in $R^n$ \begin{equation} \left\{\begin{array}{ll} (-\Delta)^{\alpha/2} u_i (x)= f_i( u_1(x), \cdots u_m(x)), & i=1, \cdots, m, \\ u_i \geq 0 , & i=1, \cdots, m, （1） \end{array} \right. \label{b1} \end{equation} where $\alpha$ is any real number between $0$ and $2$.

We obtain radial symmetry in the critical case and non-existence in the subcritical case for positive solutions.

To this end, we first establish the equivalence between (1) and the corresponding integral system $$ \left\{\begin{array}{ll} u_i(x) = \int_{R^n} \frac{c_n}{|x-y|^{n-\alpha}} f_i( u_1(y), \cdots, u_m(y)), & i=1, \cdots, m, \\ u_i(x) \geq 0, & i=1, \cdots, m. \end{array} \right. $$ A new idea is introduced in the proof, which may hopefully be applied to many other problems. Combining this equivalence with the existing results on the integral system, we obtained much more general results on the qualitative properties of the solutions for (1).

2019 Impact Factor: 1.338

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