An elliptic population system with multiple functions

Survivals of two cooperating species of animals

In this paper, we investigate the effects of perturbations on the coexistence state of the general competition model for multiple species. Previous work by Kang, Lee and Oh (see [11]) established sufficient conditions for the uniqueness of the positive solution to the following general elliptic system for multiple competing species of animals: \[\Delta u_{i} +u_{i}g_{i}(u_{1},u_{2},\ldots ,u_{i},u_{i+1},\ldots ,u_{N}) =0 \ \mbox{in} \ \Omega, \ u_{i}|_{\partial\Omega} = 0\]for $i = 1,\ldots ,N$. That is, they proved that under certain conditions, the species can coexist and that the coexistence state is unique at fixed rates. In this paper, we extend their uniqueness results by perturbing functions $g_{i}$'s of the above model, and applying super-sub solutions, maximum principles and spectrum estimates. Our arguments also rely on some detailed properties for the solution of logistic equations. By applying these techniques, we obtain sufficient conditions for the existence and uniqueness of a time independent coexistence state for the perturbed general competition model.

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.

summary:The existence of a positive solution for the generalized predator-prey model for two species $$ \begin{gathered} \Delta u + u(a + g(u,v)) = 0\quad \mbox {in}\ \Omega ,\\ \Delta v + v(d + h(u,v)) = 0\quad \mbox {in} \ \Omega ,\\ u = v = 0\quad \mbox {on}\ \partial \Omega , \end{gathered} $$ are investigated. The techniques used in the paper are the elliptic theory, upper-lower solutions, maximum principles and spectrum estimates. The arguments also rely on some detailed properties of the solution of logistic equations.

In this article, we consider some critical Brézis-Nirenberg problems in dimension $N \geq 3$ that do not have a solution. We prove that a supercritical perturbation can lead to the existence of a positive solution. More precisely, we consider the equation: \begin{equation*} \left\{ \begin{array}{rllll} -\Delta u & = & \lambda u^{q-1} + u^{2^*+ r^\alpha -1} & \mbox{in} & B, \\ u & \gt & 0 & \mbox{in} & B, \\ u&=&0 & \mbox{on} & \partial B,\\ \end{array} \right. \end{equation*} where $B \subset \mathbb{R}^N$ is a unit ball centred at the origin, $N\geq 3$ , $r=\vert x \vert$ , $\alpha \in (0,\min\{N/2,N-2\})$ , λ is a fixed real parameter and $q\in [2,2^*]$ . This class of problems can be interpreted as a perturbation of the classical Brézis–Nirenberg problem by the term r α at the exponent, making the problem supercritical when $r \in (0,1)$ . More specifically, we study the effect of this supercritical perturbation on the existence of solutions. In particular, when N = 3, an interesting and unexpected phenomenon occurs. We obtain the existence of solutions for λ in a range where the Brézis–Nirenberg problem has no solution.

This paper is devoted to study the symmetry and monotonicity of positive solutions for linear coupling elliptic systems in a ball in R N . Using the Alexandrov–Serrin method of moving planes combined with the strong maximum principle, we prove that the solutions of elliptic systems with linear couplings in a ball are symmetric w.r.t. 0 and radially decreasing. For our problems, the tangential gradient of solutions and the coupling conditions play important roles in using the moving plane method. Our results on the symmetry of solutions are further research based on the existence of solutions in [1].

and Applied Analysis 3 1-homogeneous operator in a Banach space and then demonstrate its application in establishing the existence of positive solutions for p-Laplacian boundary value problems under certain conditions. (xi) In the paper titled “Existence of solutions for nonhomogeneous A-harmonic equations with variable growth,” the authors establish a theorem for the existence of weak solutions for nonhomogeneous A-harmonic equations in subspace and then give three examples to demonstrate its application. (xii) In the paper titled “Multiple solutions for degenerate elliptic systems near resonance at higher eigenvalues,” the authors study the degenerate semilinear elliptic system in an open bounded domain with smooth boundary, and some multiplicity results of solutions are obtained for the system near resonance at certain eigenvalues by the classical saddle point theorem and a local saddle point theorem in critical point theory. (xiii) In the paper titled “A regularity criterion for the Navier-Stokes equations in the multiplier spaces,” the authors establish a regularity criterion in terms of the pressure gradient for weak solutions to the NavierStokes equations in a special class. The third set of papers, including four papers, deal with several boundary value problems for highly nonlinear ordinary differential equations. (i) In the paper titled “Positive solutions for second-order singular semipositone differential equations involving Stieltjes integral conditions,” the authors investigate the existence of positive solutions for second-order singular differential equations with a negatively perturbed term, by means of the fixed-point theory in cones. (ii) In the paper titled “Positive solutions for Sturm-Liouville boundary value problems in a Banach Space,” the sufficient conditions for the existence of single and multiple positive solutions for a second-order SturmLiouville boundary value problem are established in a Banach space, by using the fixed-point theorem of strict set contraction operators in the frame of the ODE technique. (iii) In the paper titled “Positive solutions of a nonlinear fourth-order dynamic eigenvalue problem on time scales,” the authors study a nonlinear fourth-order dynamic eigenvalue problem on time scales and obtain the existence and nonexistence of positive solutions when 0 λ, respectively, for some λ, by using the Schauder fixed-point theorem and the upper and lower solution method. (iv) In the paper titled “Bifurcation analysis for a predatorprey model with time delay and delay-dependent parameters,” a class of stage-structured predator-prey model with time delay and delay-dependent parameters is considered. By using the normal form theory and center manifold theory, some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifur-cating from Hopf bifurcations are obtained. The fourth set of papers focus on finding the approximate and numerical solutions of various complex nonlinear boundary value problems. (i) In the paper titled “On spectral homotopy analysis method for solving linear Volterra and Fredholm integrodifferential equations,” a spectral homotopy analysis method (SHAM) is proposed to solve linear Volterra integrodifferential equations, and some examples are given to test the efficiency and the accuracy of the proposed method. (ii) In the paper titled “The solution of a class of singularly perturbed two-point boundary value problems by the iterative reproducing kernel method,” the authors establish an iterative reproducing kernel method (IRKM) for solving singular perturbation problems with boundary layers and give two numerical examples to demonstrate the effectiveness of the method. (iii) In the paper titled “A Galerkin solution for Burgers’ equation using cubic B-spline finite elements,” a Galerkin method using cubic B-splines is set up to find the numerical solutions of Burgers’ equation, and the method is shown to be capable of solving Burgers’ equation accurately for values of viscosity ranging from very small to very large. (iv) In the paper titled “Forward-backward splitting methods for accretive operators in Banach spaces,” the authors introduce two iterative forward-backward splitting methods with relaxations to find zeros of the sum of two accretive operators in Banach spaces and prove the weak and strong convergence of these methods under mild conditions, and also discuss applications of these methods to variational inequalities, the split feasibility problem, and a constrained convex minimization problem. Yong Hong Wu Lishan Liu Benchawan Wiwatanapataphee Shaoyong Lai Submit your manuscripts at http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematics Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematical Problems in Engineering Hindawi Publishing Corporation http://www.hindawi.com Differential Equations International Journal of Volume 2014 Applied Mathematics Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Probability and Statistics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematical Physics Advances in Complex Analysis Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Optimization Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Combinatorics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Operations Research Advances in

On the existence of weak solutions of semilinear elliptic equations and systems with Hardy potentials

We analyse the dynamics of a prototype model for competing species with diffusion coefficients (d1d2) in a heterogeneous environment Ω. When diffusion is switched off, at each pointx∊ Ω we have a pair of ODE's: thekinetic. If for somex∊ Ω kinetic has a unique stable coexistence state, we show that there existsuch that for everythe RD-model ispersistent, in the sense that it has a compact global attractor within the interior of the positive cone and has a stable coexistence state. The same result is true if there existxu,xv∊ Ω such that the semitrivial coexistence states (u, 0) and (0,v) of thekineticare globally asymptotically stable atx=xuandx=xv, respectively. More generally, our main result shows that, for most kinetic patterns, stable coexistence of xspopulations can be found for some range of the diffusion coefficients.Singular perturbation techniques, monotone schemes, fixed point index, global analysis ofpersistence curves, global continuation and singularity theory are some of the technical tools employed to get the previous results, among others. These techniques give us necessary and/or sufficient conditions for the existence and uniqueness of coexistence states, conditions which can be explicitly evaluated by estimating some principal eigenvalues of certain elliptic operators whose coefficients are solutions of semilinear boundary value problems.We also discuss counterexamples to the necessity of the sufficient conditions through the analysis of the local bifurcations from the semitrivial coexistence states at the principal eigenvalues. An easy consequence of our analysis is the existence of models having exactly two coexistence states, one of them stable and the other one unstable. We find that there are also cases for which the model hasthree or morecoexistence states.

In biological models, the equilibrium solution is a bounded positive solution, which has very important practical significance. In this paper, a class of nonlinear delay difference equations is proposed, which is applicable to a variety of common biological population models, such as Nicholson’s blowflies and hematopoietic stem cell models. This article is to discuss this kind of nonlinear delay difference equation boundedness of the solution of the equilibrium, in the existing literature, this paper studies the nonlinear delay difference equation of the existence of bounded positive solutions of thought method, using structure function method and reduction to absurdity proved the existence of positive solutions of the equation, a class of nonlinear delay difference equation is obtained sufficient conditions for the existence of bounded positive solutions of, and proof is given. It is proved that the equilibrium solution of the nonlinear delay difference equation is always a bounded positive solution if the function f(x) is monotonically decreasing function or unimodal function and the function g(x) is monotonically increasing function. The relevant results of the existing literature are generalized and improved to make the results more general.

Lagrangian coordinates in free boundary problems for parabolic equations

In the previous chapters, the major method for proving the existence of steady state solutions for elliptic systems is Theorem 1.4-2 of the type of intermediate value theorem. It essentially uses maximum principle and the homo-topic invariance of degree. Another important technique for analyzing solutions of elliptic systems is the method of monotone schemes. Besides existence, it can be adapted to study uniqueness and stability for corresponding parabolic systems. Moreover, an analogous theory can be developed for finite difference systems. The corresponding monotone schemes provide numerical method for studying elliptic systems. The finite difference theory will be described in Chapter 6.

The author study the spectral properties of an eighth-order differential operator with a piecewise-smooth potential and a discontinuous weight function. For large values of the spectral parameter, the asymptotics of solutions of differential equations defining the operator under study is studied. With the help of the obtained asymptotics, the conditions of “conjugation” at the point of discontinuity of the coefficients, the necessity of which follows from physical considerations, are studied. The separated boundary conditions that define the operator are studied. An indicator diagram of an equation whose roots are the eigenvalues of the operator is investigated. The asymptotics of the eigenvalues of the differential operator under study is found. Using the Lidskyi - Sadovnichyi method, the first regularized trace of the differential operator is calculated.

The purpose of this paper is to give conditions for the uniqueness of positive solution to a rather general type of elliptic system of the Dirichlet problem on a bounded domain &#8486; in <i>R</i>^<i>n</i>. Also considered are the effects of perturbations on the coexistence state and uniqueness.

In this paper, we concentrate on the uniqueness of the positive solution for the general elliptic system 8 : ¢u + u(g1(u) i g2(v)) = 0 ¢v + v(h1(u) i h2(v)) = 0 in R + £ ›; uj@› = vj@› = 0: This system is the general model for the steady state of a com- petitive interacting system. The techniques used in this paper are upper-lower solutions, maximum principles and spectrum estimates. The arguments also rely on some detailed properties for the solution of logistic equations.