Cooperative Elliptic Systems Research Articles

Vanishing property of principal eigenfunction for cooperative elliptic systems

Vanishing property of principal eigenfunction for cooperative elliptic systems

Existence and non-existence results for cooperative elliptic systems without variational structure

Existence and non-existence results for cooperative elliptic systems without variational structure

On the monotonicity property of the generalized eigenvalue for weakly-coupled cooperative elliptic systems

We consider general linear non-degenerate weak-coupled cooperative elliptic systems and study certain monotonicity properties of the generalized principal eigenvalue in Rd with respect to the potential. It is shown that monotonicity on the right is equivalent to the recurrence property of the twisted operator which is, in turn, equivalent to the minimal growth property at infinity of the principal eigenfunctions. The strict monotonicity property of the principal eigenvalue is shown to be equivalent with the exponential stability of the twisted operators. An equivalence between the monotonicity property on the right and the stochastic representation of the principal eigenfunction is also established.

Strong Maximum Principle for Viscosity Solutions of Fully Nonlinear Cooperative Elliptic Systems

In this paper, we consider the validity of the strong maximum principle for weakly coupled, degenerate and cooperative elliptic systems in a bounded domain. In particular, we are interested in the viscosity solutions of elliptic systems with fully nonlinear degenerated principal symbol. Applying the method of viscosity solutions, introduced by Crandall, Ishii and Lions in 1992, we prove the validity of strong interior and boundary maximum principle for semi-continuous viscosity sub- and super-solutions of such nonlinear systems. For the first time in the literature, the strong maximum principle is considered for viscosity solutions to nonlinear elliptic systems. As a consequence of the strong interior maximum principle, we derive comparison principle for viscosity sub- and super-solutions in case when on of them is a classical one. The main novelty of this work is the reduction of the smoothness of the solution. In the literature the strong maximum principle is proved for classical C2 or generalized C1 solutions, while we prove it for semi-continuous ones.

Solutions to indefinite weakly coupled cooperative elliptic systems

We study the elliptic system \begin{equation*} \begin{cases} -\Delta u_1 - \kappa_1u_1 = \mu_1|u_1|^{p-2}u_1 + \lambda\alpha|u_1|^{\alpha-2}|u_2|^\beta u_1, \\ -\Delta u_2 - \kappa_2u_2 = \mu_2|u_2|^{p-2}u_2 + \lambda\beta|u_1|^\alpha|u_2|^{\beta-2}u_2, \\ u_1,u_2\in D^{1,2}_0(\Omega), \end{cases} \end{equation*} where $\Omega$ is a bounded domain in $\mathbb R^N$, $N\geq 3$, $\kappa_1,\kappa_2\in\mathbb R$, $\mu_1,\mu_2,\lambda> 0$, $\alpha,\beta> 1$, and $\alpha + \beta = p\le 2^*:={2N}/({N-2})$. For $p\in (2,2^*)$ we establish the existence of a ground state and of a prescribed number of fully nontrivial solutions to this system for $\lambda$ sufficiently large. If $p=2^*$ and $\kappa_1,\kappa_2> 0$ we establish the existence of a ground state for $\lambda$ sufficiently large if, either $N\ge5$, or $N=4$ and neither $\kappa_1$ nor $\kappa_2$ are Dirichlet eigenvalues of $-\Delta$ in $\Omega$.

Boundary Control for Cooperative Elliptic Systems under Conjugation Conditions

The existence and uniqueness of the state for 2 × 2 Dirichlet cooperative elliptic systems under conjugation conditions are proved using Lax-Milgram lemma, then the boundary control for these systems is discussed. The set of equations and inequalities that characterizes this boundary control is found by theory of Lions, Sergienko and Deineka. The problem for cooperative Neumann elliptic systems under conjugation conditions is also considered. Finally, the problem for n × n cooperative elliptic systems under conjugation conditions is established.

A symmetry result for cooperative elliptic systems with singularities

We obtain symmetry results for solutions of an elliptic system of equationpossessing a cooperative structure. The domain in which the problem is set maypossess holes or small (measured in terms of capacity) along which thesolution may diverge.The method of proof relies on the moving plane technique, which needs to besuitably adapted here to take care of the complications arising from the vacancies inthe domain and the analytic structure of the elliptic system.

Symmetry results of large solutions for semilinear cooperative elliptic systems

In this paper, we consider symmetry of large solutions of the semilinear cooperative elliptic system Δui=gi(u1,...,uM) in BR,ui=+∞, on ∂BR, where ui∈C2(BR),gi:RM→R a C function, BR is the open ball of center 0 and radius R>0 in RN,N≥3,M∈N,i=1,2,...,M. After introducing the weakly coupled, fully coupled hypotheses, and some assumptions for g, we use the moving plane method to prove any large solution of the semilinear cooperative elliptic system is radially symmetric and radially increasing. In order to use the moving plane method for our problem, a strong maximum principle of the linear cooperative elliptic system is constructed. Furthermore, we also get the symmetry result of large solutions of the semilinear cooperative elliptic system in an annulus.

Symmetry and monotonicity properties of singular solutions to some cooperative semilinear elliptic systems involving critical nonlinearities

We investigate qualitative properties of positive singular solutions of some elliptic systems in bounded and unbounded domains. We deduce symmetry and monotonicity properties via the moving plane procedure. Moreover, in the unbounded case, we study some cooperative elliptic systems involving critical nonlinearities in \begin{document}$ {\mathbb{R}}^n $\end{document} .

A spatial SEIRS reaction-diffusion model in heterogeneous environment

We propose a susceptible-exposed-infected-recovered-susceptible (SEIRS) reaction-diffusion model, where the disease transmission and recovery rates can be spatially heterogeneous. The basic reproduction number (R0) is connected with the principal eigenvalue of a linear cooperative elliptic system. Threshold-type results on the global dynamics in terms of R0 are established. The monotonicity of R0 with respect to the diffusion rates of the exposed and infected individuals, which does not hold in general, is established in several cases. Finally, the asymptotic profile of the endemic equilibrium is investigated when the diffusion rate of the susceptible individuals is small. Our results reveal the importance of the movement of the exposed and recovered individuals in disease dynamics, as opposed to most of previous works which solely focused on the movement of the susceptible and infected individuals.

DISTRIBUTED CONTROL FOR COOPERATIVE ELLIPTIC SYSTEMS UNDER CONJUGATION CONDITIONS

The cooperative elliptic systems involving Laplace operator defined on bounded, continuous and strictly Lipschitz domains ofunder conjugation conditions are considered. First, the existence and uniqueness of the state for these systems with Dirichlet and conjugation conditions is proved, then the set of equations and inequalities that characterizes the distributed control of these systems is found. The problem with Neumann conditions is also discussed.

Existence results of positive solutions for nonlinear cooperative elliptic systems involving fractional Laplacian

In this paper, we prove existence results of positive solutions for the following nonlinear elliptic problem with gradient terms: [Formula: see text] where [Formula: see text] denotes the fractional Laplacian and [Formula: see text] is a smooth bounded domain in [Formula: see text]. It shown that under some assumptions on [Formula: see text] and [Formula: see text], the problem has at least one positive solution [Formula: see text]. Our proof is based on the classical scaling method of Gidas and Spruck and topological degree theory.

Multiple solutions of superlinear cooperative elliptic systems at resonant

In this paper, we study a class of resonant cooperative elliptic systems with nonlinearities being superlinear at infinity. We remove some classical conditions near 0 used before by many authors, and we obtain infinitely many nontrivial solutions for this class of systems, a topic about which very little is known. Our main result extends and improves some recent results in the literature.

OPTIMAL DISTRIBUTED CONTROL OF STOCHASTIC ELLIPTIC SYSTEMS WITH CONSTRAINTS

The objective of this paper is to study the optimality for stochastic non cooperative elliptic systems. A distributed control problem for a stochastic elliptic systems with constraints on states and controls is studied. First, the existence and uniqueness of the state process for these systems are proved. The necessary and sufficient conditions of optimality are derived for the Dirichlet and Neumann problems.

A maximum principle for some nonlinear cooperative elliptic PDE systems with mixed boundary conditions

One of the classical maximum principles states that any nonnegative solution of a proper elliptic PDE attains its maximum on the boundary of a bounded domain. We suitably extend this principle to nonlinear cooperative elliptic systems with diagonally dominant coupling and with mixed boundary conditions. One of the consequences is a preservation of nonpositivity, i.e. if the coordinate functions or their fluxes are nonpositive on the Dirichlet or Neumann boundaries, respectively, then they are all nonpositive on the whole domain as well. Such a result essentially expresses that the studied PDE system is a qualitatively reliable model of the underlying real phenomena, such as proper reaction-diffusion systems in chemistry.

MULTIPLICITY RESULT OF THE SOLUTIONS FOR A CLASS OF THE ELLIPTIC SYSTEMS WITH SUBCRITICAL SOBOLEV EXPONENTS

This paper is devoted to investigate the multiple solutions for a class of the cooperative elliptic system involving subcritical Sobolev exponents on the bounded domain with smooth boundary. We first show the uniqueness and the negativity of the solution for the linear system of the problem via the direct calculation. We next use the variational method and the mountain pass theorem in the critical point theory.

Ground State Solutions for Resonant Cooperative Elliptic Systems with General Superlinear Terms

This paper is concerned with the following cooperative elliptic system: $$\left\{ \begin{array}{ll} -\Delta u=\xi u+f(x,u,v), \quad \quad \mbox{in } \Omega, -\Delta v=\zeta v+g(x,u,v), \quad \quad \mbox{in } \Omega, u=v=0, \quad \quad\mbox{on } \partial\Omega, \end{array} \right.$$ where \({U=(u,v): \Omega\rightarrow \mathbb{R}^{2}}\), Ω is a bounded smooth domain in \({\mathbb{R}^{N}}\) and \({\xi,\zeta\in\mathbb{R}}\). We establish the existence of ground state solutions for this system using a much more direct approach to find a minimizing Cerami sequence for the energy functional outside the generalized Nehari manifold developed recently by Szulkin and Weth.

Asymptotic Behavior of the Principal Eigenvalue for Cooperative Elliptic Systems and Applications

The asymptotic behavior of the principal eigenvalue for general linear cooperative elliptic systems with small diffusion rates is determined. As an application, we show that if a cooperative system of ordinary differential equations has a unique positive equilibrium which is globally asymptotically stable, then the corresponding reaction-diffusion system with either the Neumann boundary condition or the Robin boundary condition also has a unique positive steady state which is globally asymptotically stable, provided that the diffusion coefficients are sufficiently small. Moreover, as the diffusion coefficients approach zero, the positive steady state of the reaction-diffusion system converges uniformly to the equilibrium of the corresponding kinetic system.

Estimate of Morse index of cooperative elliptic systems and its application to spatial vector solitons

Local instability index of unstable solutions to single partial differential equations (PDEs) by a local minimax method (LMMM) was established in Zhou (2005). It is known that the local min-orthogonal method (LMOM) which was first proposed in Zhou (2004) and then further developed in Chen et al. (2008) can find more general unstable solutions to both single PDEs and cooperative elliptic systems. This paper is to carry out instability analysis of unstable solutions by LMOM, to which an infinite-dimensional functional space can be decomposed as a direct sum of a finite-dimensional subspace and its orthogonal complement. A Morse index approach is developed to show that with LMOM, instability behavior of a solution in such infinite-dimensional complement subspace can be totally determined. Usual instability analysis in an entire space is then reduced to analysis in its finite-dimensional subspace, for which a corresponding matrix decomposition is proposed to analyze a solution’s instability behavior. Estimates of Morse index are also established. Finally, numerical examples of both 2- and 3-component cooperative systems arising in nonlinear optics are carried out for spatial vector solitons, whose local instabilities are numerically confirmed by the new estimates. Certain important properties of the examples are also verified or presented.

GROUND STATE SOLUTIONS OF NON-RESONANT COOPERATIVE ELLIPTIC SYSTEMS WITH SUPERLINEAR TERMS

In this paper, we study the existence of ground state solutions for a class of non-resonant cooperative elliptic systems by a variant weak linking theorem. Here the classical Ambrosetti-Rabinowitz superquadratic condition is replaced by a general super quadratic condition.