Asymptotic Behavior Of Positive Solutions Research Articles

Asymptotic Behavior of Positive Solutions for Three Types of Fractional Difference Equations with Forcing Term

In this paper, we consider three types of fractional difference equations with forcing term. The asymptotic behaviors of positive solutions for the proposed equations are extensively investigated. Our approach mainly relies on the features of discrete fractional calculus and some mathematical inequalities. To conclude our theoretical findings, we give an example demonstrating complete consistency to the main results.

Asymptotic behavior of positive solutions to a nonlinear biharmonic equation near isolated singularities

In this paper, we consider the asymptotic behavior of positive solutions of the biharmonic equation $$\begin{aligned} \Delta ^2 u = u^p\quad \text{ in }\; B_1\backslash \{0\} \end{aligned}$$with an isolated singularity, where the punctured ball \(B_1 \backslash \{0\} \subset {\mathbb {R}}^n\) with \(n\ge 5\) and \(\frac{n}{n-4}< p < \frac{n+4}{n-4}\). This equation is relevant for the Q-curvature problem in conformal geometry. We classify isolated singularities of positive solutions and describe the asymptotic behavior of positive singular solutions without the sign assumption for \(-\Delta u\). We also give a new method to prove removable singularity theorem for nonlinear higher order equations.

Existence and asymptotic behavior of positive solutions for Kirchhoff type problems with steep potential well

In this paper, we consider the following Kirchhoff type problem{−(a+b∫R3|∇u|2dx)Δu+λV(x)u=|u|p−2uinR3,u∈H1(R3), where a>0 is a constant, b and λ are positive parameters, and 2<p<6. Suppose that the nonnegative continuous potential V represents a potential well with the bottom V−1(0), the equation has been extensively studied in the case 4≤p<6. In contrast, no existence result of solutions is available for the case 2<p<4 due to the presence of the term (∫R3|∇u|2dx)Δu. By combining the truncation technique and the parameter-dependent compactness lemma, we prove the existence of positive solutions for b small and λ large in the case 2<p<4. Moreover, we also explore the decay rate of the positive solutions as |x|→∞ as well as their asymptotic behavior as b→0 and λ→∞.

Asymptotic behavior at the isolated singularities of solutions of some equations on singular manifolds with conical metrics

We present the sharp characterization of the behavior at the isolated singularities of positive solutions of some equations on singular manifolds with conical metrics. It is seen that the equations on singular manifolds with conical metrics are equivalent to weighted elliptic equations in where is the unit ball. The weights can be singular at x = 0. We present the sharp asymptotic behavior of positive solutions of the weighted elliptic equations at x = 0 and establish expansions of these solutions up to arbitrary orders. Asymptotic behavior at the isolated singularitie of positive solutions of elliptic equations without weights has been studied by many authors. We will obtain new results on the asymptotic behavior at the isolated singularities even for positive solutions of equations without weights in the subcritical case.

Existence and asymptotic behaviour of positive solutions to a stochastic multispecies Holling type II model

In this paper, we develop a stochastic Holling type II model from the commonly used deterministic Holling type II model by introducing Brownian motion as a stochastic perturbation, which can naturally account for the external environmental effects on a multispecies community. We study the existence and uniqueness of positive solutions to the stochastic model, and we establish sufficient conditions for strong persistence and extinction.

Influence of singular weights on the asymptotic behavior of positive solutions for classes of quasilinear equations

Main objective of this paper is to study positive decaying solutions for a class of quasilinear problems with weights. We consider one dimensional problems on an interval which may be finite or infinite. In particular, when the interval is infinite, unlike the known cases in the history where constant weights force the solution not to decay, we discuss singular weights in the diffusion and reaction terms which produce positive solutions that decay to zero at infinity. We also discuss singular weights that lead to positive solutions not satisfying Hopf’s boundary lemma. Further, we apply our results to radially symmetric solutions to classes of problems in higher dimensions, say in an annular domain or in the exterior region of a ball. Finally, we provide examples to illustrate our results.

On a class of nonlocal nonlinear Schrödinger equations with potential well

Abstract In this paper we investigate the existence, multiplicity and asymptotic behavior of positive solution for the nonlocal nonlinear Schrödinger equations. We exploiting the relationship between the Nehari manifold and eigenvalue problems to discuss how the Nehari manifold changes as parameters μ, λ changes and show how existence, multiplicity and asymptotic results for positive solutions of the equation are linked to properties of the manifold.

Asymptotic behavior of positive solutions to a predator–prey elliptic system with strong hunting cooperation in predators

In predator–prey interactions, predators often collaborate when hunting their prey. We study a predator–prey elliptic system reflecting such a mechanism. Mainly, we obtain the asymptotic behavior of positive solutions to the system when the cooperation effect of the predators is strong.

Existence and asymptotic properties of positive solutions for a general quasilinear Schr\xf6dinger equation

By a change of variables with cut-off functions, we study the existence and the asymptotic behavior of positive solutions for a general quasilinear Schrödinger equation which arises from plasma physics. We extend the results of (Adv. Nonlinear Stud. 18(1):131-150, 2017) from alpha=1 to alpha>frac{1}{2}. Especially, we can consider the exponent p in (2,2^{*}) for all Ngeq3.

Existence and exact asymptotic behaviour of positive solutions for fractional boundary value problem with P-Laplacian operator

ABSTRACTThis paper deals with existence, uniqueness and global behaviour of a positive solution for the fractional boundary value problem in with the condition where , , p>1, , the differential operator is taken in the Riemann–Liouville sense and are non-negative and continuous functions that may are singular at x=0 or x=1 and satisfies some appropriate conditions.

Asymptotic behavior of positive solutions of nonlinear fractional differential equations with Caputo-type Hadamard derivative

Asymptotic behavior of positive solutions of nonlinear fractional differential equations with Caputo-type Hadamard derivative

On some difference equations of exponential form

In this paper, the local asymptotic behavior of positive solutions of some exponential difference equations x_(n+1)=(x_n+x_(n-k))/(1+x_(n-k) e^(x_(n-k) ) ) , k ∈ N, n=0,1,2,… are investigated where the initial conditions are arbitrary positive real numbers. Furthermore, some numerical examples are presented to verify our results.

Asymptotic behavior of positive solutions of the Henon equation

We investigate the radial positive solutions of the Henon equation. It is known that this equation has three different types of radial solutions: the M-solutions (singular at $r=0$), the E-solutions (regular at $r=0$) and the F-solutions (whose existence begins away from $r=0$). For the M-solutions and E-solutions, by virtue of some prior estimates, we adopt a circulating iterative method, step-by-step, to derive their precise asymptotic expansions. In particular, the M-solution has an extremely plentiful structure, and its asymptotic expansions are more complicated. In contrast to previous research Bratt and Pfaffelmoser, and Gidas and Spruck, our results are more accurate.

Concentration and energy invariance for a class of fully nonlinear elliptic equations

We study the asymptotic behavior of positive solutions of fully nonlinear elliptic equations in a ball, as the exponent of the power nonlinearity approaches a critical value. We show that solutions concentrate and blow up at the center of the ball, while a suitable associated energy remains invariant.

Positive steady states for a prey–predator model with population flux by attractive transition

This paper studies the stationary solutions of a prey–predator model with population flux by attractive transition. We first obtain a bifurcation branch (connected set) of positive solutions which connects two semitrivial solutions. Next we derive the asymptotic behavior of positive solutions as the coefficient α of the population flux tends to infinity. A main result implies that positive solutions can be classified into two types as α→∞. In one type of them, as α→∞, positive solutions of the prey–predator model approach positive solutions of a competition model with equal diffusion coefficients.

Asymptotic behavior and stability of positive solutions to a spatially heterogeneous predator–prey system

In this paper, we continue to study a spatially heterogeneous predator–prey system where the interaction is governed by a Holling type II functional response, which has been studied in Du and Shi (2007) [14]. We further study the asymptotic profile of positive solutions and give a complete understanding of coexistence region. Moreover, a good understanding of the number, stability and asymptotic behavior of positive solutions is gained for large m. Finally, we further compare the difference of steady-state solutions between m>0 and m=0. It turns out that the spatial heterogeneity of the environment and the Holling type II functional response play a very important role in this model.

Positive solutions of a predator-prey model with cross-diffusion

In this paper, we consider the positive solutions for a predator-prey model with cross-diffusion and Holling type II functional response. In particular, the existence of positive solutions can be established by the bifurcation theory. Moreover, the uniqueness and the exact number of positive solutions is studied when the parameter m is large. Furthermore, for large cross-diffusion rate α with the spatial dimension is less than 5, we can derive the corresponding limit systems and also study the asymptotic behavior of positive solutions. Finally, some numerical simulations are presented to supplement the analysis results in one dimension case. This results give us some important information on the structure of positive solutions.

Asymptotic Behavior of Global Positive Solution to an Information Diffusion Model with Random Perturbation in Social Network

The paper explore an information diffusion models with random perturbation in social network. First, we show the models exit the unique global positive solution. By the construction of the Lyapunov function, we give the positive solution is stochastically asymptotically stable in the large around disease-free equilibrium, i.e. the conditions of the information diffusion will die out, investigate the stochastic asymptotic behavior of the positive solution around endemic equilibrium of the deterministic models, obtain the stochastic asymptotic stability condition, i.e. the conditions of the information diffusion will be persistent in social networks.

Positive solutions of singular semilinear elliptic problems in NTA-cones

We study the existence, the uniqueness and the asymptotic behavior of positive solutions of the nonlinear equation \Delta v+ f(.,v)=0, in an NTA-cone in \mathbb R^n( n\geq 3 ), when a positive Borel measurable function f(.,.) is continuous and non-increasing with respect to the second variable and satisfies a certain condition related to a Kato class.

Regularly varying sequences and Emden–Fowler type second-order difference equations

We study asymptotic behavior of positive solutions of nonlinear difference equation where are regularly varying sequences. It is shown that with the help of discrete regular variation, complete information can be acquired about the existence of all possible strongly increasing regularly varying solutions of this equation and their accurate asymptotic behavior at infinity.