Asymptotic Behavior Of Positive Solutions Research Articles

An asymptotic behavior of positive solutions for a new class of elliptic systems involving of $$\left( p\left( x\right) ,q\left( x\right) \right) $$ p x , q x -Laplacian systems

The propose of this paper is to study of the existence and asymptotic behavior of positive solutions for a new class of elliptic systems involving of $$\left( p\left( x\right) ,q\left( x\right) \right) $$ -Laplacian systems using sub-super solutions method, with respect to the symmetry conditions. Our results are natural extensions from the previous recent ones in Edmunds and Rakosnk (Proc R Soc Lond Ser A 437:229–236, 1992).

Existence and Asymptotic Behavior of Positive Solutions for a Class of Quasilinear Schrödinger Equations

Abstract In this paper, we study the quasilinear Schrödinger equation - Δ ⁢ u + V ⁢ ( x ) ⁢ u - γ 2 ⁢ ( Δ ⁢ u 2 ) ⁢ u = | u | p - 2 ⁢ u {-\Delta u+V(x)u-\frac{\gamma}{2}(\Delta u^{2})u=|u|^{p-2}u} , x ∈ ℝ N {x\in\mathbb{R}^{N}} , where V ⁢ ( x ) : ℝ N → ℝ {V(x):\mathbb{R}^{N}\to\mathbb{R}} is a given potential, γ > 0 {\gamma>0} , and either p ∈ ( 2 , 2 * ) {p\in(2,2^{*})} , 2 * = 2 ⁢ N N - 2 {2^{*}=\frac{2N}{N-2}} for N ≥ 4 {N\geq 4} or p ∈ ( 2 , 4 ) {p\in(2,4)} for N = 3 {N=3} . If γ ∈ ( 0 , γ 0 ) {\gamma\in(0,\gamma_{0})} for some γ 0 > 0 {\gamma_{0}>0} , we establish the existence of a positive solution u γ {u_{\gamma}} satisfying max x ∈ ℝ N ⁡ | γ μ ⁢ u γ ⁢ ( x ) | → 0 {\max_{x\in\mathbb{R}^{N}}|\gamma^{\mu}u_{\gamma}(x)|\to 0} as γ → 0 + {\gamma\to 0^{+}} for any μ > 1 2 {\mu>\frac{1}{2}} . Particularly, if V ⁢ ( x ) = λ > 0 {V(x)=\lambda>0} , we prove the existence of a positive classical radial solution u γ {u_{\gamma}} and up to a subsequence, u γ → u 0 {u_{\gamma}\to u_{0}} in H 2 ⁢ ( ℝ N ) ∩ C 2 ⁢ ( ℝ N ) {H^{2}(\mathbb{R}^{N})\cap C^{2}(\mathbb{R}^{N})} as γ → 0 + {\gamma\to 0^{+}} , where u 0 {u_{0}} is the ground state of the problem - Δ ⁢ u + λ ⁢ u = | u | p - 2 ⁢ u {-\Delta u+\lambda u=|u|^{p-2}u} , x ∈ ℝ N {x\in\mathbb{R}^{N}} .

Effect of cross-diffusion on the stationary problem of a Leslie prey-predator model with a protection zone

In this work, we study the change of behavior of positive solutions in a Leslie predator-prey model when a simple protection zone and cross-diffusion for the prey are introduced. We analyze the effects of cross-diffusion and protection zone on the bifurcation continuum of positive solutions. The asymptotic behavior of positive solutions is also discussed as the cross-diffusion and the birth rate of the predator tend to infinity, respectively. Finally, for small birth rates of two species and large cross-diffusion for the prey, the detailed structure and stability of positive solutions are established. Our results indicate that the environmental heterogeneity, together with large cross-diffusion, can produce much more complicated stationary patterns, moreover, our research here reveals significant difference from those studied in Du et al. (J Differ Equ 246:3932-3956, 2009), Oeda (J Differ Equ 250:3988-4009, 2011) and Wang and Li (Nonlinear Anal Real World Appl 14:224-245, 2013).

Existence and Asymptotic Behavior of Positive Solutions for a Coupled Fractional Differential System

In this paper, we take up the existence and the asymptotic behavior of positive and continuous solutions to the following coupled fractional differential system $$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle D^{\alpha } u=a(x)\displaystyle u^{p }\displaystyle v^{r}\quad \text { in }(0,1) , \\ \displaystyle D^{\beta } v=b(x)\displaystyle u^{s }\displaystyle v^{q}\quad \text { in }(0,1) , \\ u(0)= u(1)= D^{\alpha -3}u(0)= u^{\prime }(1)=0,\\ v(0)= v(1)= D^{\beta -3}v(0)= v^{\prime }(1)=0, \end{array} \right. \end{aligned}$$ where \( \alpha , \beta \in (3,4]\), \(p, q\in (-1,1)\), \(r, s\in \mathbb {R}\) such that \((1-|p|)(1-|q|)-|rs|> 0\), D is the standard Riemann–Liouville differentiation and a, b are nonnegative and continuous functions in (0, 1) allowed to be singular at \(x=0\) and \(x=1\) and they are required to satisfy some appropriate conditions related to Karamata regular variation theory.

Existence and asymptotic behaviour of positive solutions for semilinear polyharmonic coupled systems

Let m be a positive integer. We investigate the existence and the asymptotic behaviour of positive continuous solutions to the following coupled polyharmonic system in the unit ball B of , ,where , , r, such that and the functions a, b are nonnegative measurable on B satisfying some assumptions related to Karamata regular variation theory.

Positive solutions for Lotka–Volterra competition system with large cross-diffusion in a spatially heterogeneous environment

In this paper, we study the stationary problem for the Lotka–Volterra competition system with cross-diffusion in a spatially heterogeneous environment. Although some sufficient conditions for the existence of positive solutions are obtained by using global bifurcation theory, the information for their structure is far from complete. In order to get better understanding of the competition system with cross-diffusion, we focus on the asymptotic behaviour of positive solutions and derive two shadow systems as the cross-diffusion coefficient tends to infinity, moreover, the structure of positive solutions of the limiting system is analysed. The result of asymptotic behaviour also reveals different phenomena from that studied in Wang and Li (2013).

Asymptotic behavior of positive solutions of a Lanchester-type model

Asymptotic behavior of positive solutions of a Lanchester-type model

On the behavior of positive solutions of semilinear elliptic equations in asymptotically cylindrical domains

The goal of this note is to study the asymptotic behavior of positive solutions for a class of semilinear elliptic equations which can be realized as minimizers of their energy functionals. This class includes the Fisher-KPP and Allen–Cahn nonlinearities. We consider the asymptotic behavior in domains becoming infinite in some directions. We are in particular able to establish an exponential rate of convergence for this kind of problems.

Existence and global asymptotic behavior of positive solutions for superlinear fractional dirichlet problems on the half-line

Abstract Using estimates on the Green function and a perturbation argument, we prove the existence and uniqueness of a positive continuous solution to problem: { D α u ( t ) = u ( t ) φ ( t , u ( t ) ) , t ∈ ( 0 , ∞ ) , 1 < α ≤ 2 , lim t → 0 t 2 − α u ( t ) = a , lim t → ∞ t 1 − α u ( t ) = b , $$\begin{equation*} \left\{ \begin{array}{l} D^{\alpha} u(t)=u(t)\varphi (t, u(t)),\ \ t\in (0,\infty),\ 1\lt \alpha \leq 2, \\ \underset{t\rightarrow 0}{\lim} t^{2-\alpha} u(t)=a,\ \underset{ t\rightarrow \infty} {\lim} t^{1-\alpha} u(t)=b, \end{array} \right. \end{equation*}$$ where D α is the standard Riemann-Liouville fractional derivative, a, b are nonnegative constants such that a + b > 0 and φ(t, s) is a nonnegative continuous function that is required to satisfy an appropriate condition related to a class 𝜅 α satisfying suitable integrability condition. We also give a global behavior of such solution.

Qualitative properties of positive solutions to nonlocal critical problems involving the Hardy-Leray potential

We prove existence, qualitative properties and asymptotic behavior of positive solutions to the doubly critical problem $$\begin{aligned} (-\Delta )^s u=\vartheta \frac{u}{|x|^{2s}}+u^{2_s^*-1}, \quad u\in \dot{H}^s(\mathbb {R}^N), \quad N> 2s,\quad 0<s<1. \end{aligned}$$ The technique that we use to prove the existence is based on variational arguments. The qualitative properties are obtained by using the moving plane method, in a nonlocal setting, on the whole \(\mathbb {R}^{N}\) and some comparison results. Moreover, in order to find the asymptotic behavior of solutions, we use a representation result that allows to transform the original problem into a different nonlocal problem in a weighted fractional space.

Periodic-parabolic eigenvalue problems with a large parameter and degeneration

We consider a periodic-parabolic eigenvalue problem with a non-negative potential λm vanishing on a non-cylindrical domain Dm satisfying conditions similar to those for the parabolic maximum principle. We show that the limit as λ→∞ leads to a periodic-parabolic problem on Dm having a periodic-parabolic principal eigenvalue and eigenfunction which are unique in some sense. We substantially improve a result from [Du and Peng, Trans. Amer. Math. Soc. 364 (2012), p. 6039–6070]. At the same time we offer a different approach based on a periodic-parabolic initial boundary value problem. The results are motivated by an analysis of the asymptotic behaviour of positive solutions to semilinear logistic periodic-parabolic problems with temporal and spacial degeneracies.

Principal eigenvalues for some nonlocal eigenvalue problems and applications

This paper is concerned with the principal eigenvalues of some nonlocal operators. We first derive a result on the limit of certain sequences of principal eigenvalues associated with some nonlocal eigenvalue problems. Then, such a result is used to study the existence, uniqueness and asymptotic behavior of positive solutions to a nonlocal stationary problem with a parameter. Finally, the long-time behavior of the solutions of the corresponding nonlocal evolution equation and the asymptotic behavior of positive stationary solutions on parameter are discussed.

Existence and Asymptotic Behavior of Positive Solutions for a Coupled System of Semilinear Fractional Differential Equations

In this paper, we study a coupled system of semilinear Riemann–Liouville type fractional differential equations. Applying the Schauder fixed point theorem, we prove the existence of positive solutions of the fractional differential system. Furthermore, the asymptotic behavior of the solutions is given by using Karamata regular variation theory.

Existence and asymptotic behavior of positive solutions of a semilinear elliptic system in a bounded domain

Let \(\Omega\) be a bounded domain in \(\mathbb{R}^{n}\) (\(n\geq 2\)) with a smooth boundary \(\partial \Omega\). We discuss in this paper the existence and the asymptotic behavior of positive solutions of the following semilinear elliptic system \[\begin{aligned} -\Delta u&=a_{1}(x)u^{\alpha}v^{r}\quad\text{in}\;\Omega ,\;\;\,u|_{\partial\Omega}=0, -\Delta v&=a_{2}(x)v^{\beta}u^{s}\quad\text{in}\;\Omega ,\;\;\,v|_{\partial\Omega }=0.\end{aligned}\] Here \(r,s\in \mathbb{R}\), \(\alpha,\beta \lt 1\) such that \(\gamma :=(1-\alpha)(1-\beta)-rs\gt 0\) and the functions \(a_{i}\) (\(i=1,2\)) are nonnegative and satisfy some appropriate conditions with reference to Karamata regular variation theory.

GLOBAL ASYMPTOTIC BEHAVIOR OF POSITIVE SOLUTIONS FOR EXPONENTIAL FORM DIFFERENCE EQUATIONS WITH THREE PARAMETERS

In this paper, we study a class of second order difference equations with three paremeters. With positive initial values, the asymptotic behavior of positive solutions are investigated.

Note about asymptotic behaviour of positive solutions of superlinear differential equation of Emden-Fowler type at zero

We study the asymptotic behavior of solutions of the differential equation of Emden-Fowler type y'' - xayσ = 0, where a _ R, σ > 1. We prove some of theorems on asymptotic properties of solutions and obtain asymptotic formulas for solutions near zero.

Existence and global asymptotic behavior of positive solutions for sublinear and superlinear fractional boundary value problems

In this paper, the authors aim at proving two existence results of fractional differential boundary value problems of the form $$\left( {{P_{a,b}}} \right)\left\{ {_{u\left( 0 \right) = u\left( 1 \right) = 0,{D^{\alpha - 3}}u\left( 0 \right) = a,u'\left( 1 \right) = - b}^{{D^\alpha }u\left( x \right) + f\left( {x,u\left( x \right)} \right) = 0,x \in \left( {0,1} \right)}} \right.$$ , where 3 0 and f(x, t) = tφ(x, t), with φ(x, t) being a nonnegative continuous function in (0, 1)×[0,∞) that is required to satisfy some suitable integrability condition. Using estimates on the Green’s function and a perturbation argument, the authors prove the existence and uniqueness of a positive continuous solution u to problem (Pa,b), which behaves like the unique solution of the homogeneous problem corresponding to (Pa,b). Some examples are given to illustrate the existence results.

Existence and Asymptotic Behavior of Positive Solutions for Variable Exponent Elliptic Systems

AbstractIn this paper, our main purpose is to establish the existence of positive solution of the following systemwhereare constants.F(x,u,υ) = λp(x)[g(x)a(u)+f(υ)],H(x,u,υ)=θq(x)[g1(x)b(υ)+h(u)], λ,θ&gt;0 are parameters,p(x),q(x) are radial symmetric functions,is calledp(x)-Laplacian. We give the existence results and consider the asymptotic behavior of the solutions. In particular, we do not assume any symmetric condition, and we do not assume any sign condition onF(x,0,0) andH(x,0,0) either.

Boundedness and asymptotic behavior of positive solutions for difference equations of exponential form

Boundedness and asymptotic behavior of positive solutions for difference equations of exponential form

Existence and asymptotic behavior of positive solutions for a one-prey and two-competing-predators system with diffusion

A diffusive one-prey and two-competing-predators system under homogeneous Dirichlet boundary conditions is studied. First, we obtain sufficient conditions for the extinction and existence of global attractor of the time-dependent system by means of the comparison principle. Second, we discuss the existence and nonexistence of coexistence states, and give sufficient conditions for the existence of coexistence states by using the fixed point index theory. In addition, we investigate the bifurcation from a double eigenvalue by virtue of space decomposition and implicit function theorem. Finally, some numerical simulations are made to verify and complement the theoretical analysis.