Asymptotic Behavior Of Positive Solutions Research Articles

High Perturbations of a Fractional Kirchhoff Equation with Critical Nonlinearities

This paper concerns a fractional Kirchhoff equation with critical nonlinearities and a negative nonlocal term. In the case of high perturbations (large values of α, i.e., the parameter of a subcritical nonlinearity), existence results are obtained by the concentration compactness principle together with the mountain pass theorem and cut-off technique. The multiplicity of solutions are further considered with the help of the symmetric mountain pass theorem. Moreover, the nonexistence and asymptotic behavior of positive solutions are also investigated.

Existence and asymptotic behavior of positive solutions to some logarithmic Schrödinger–Poisson system

Existence and asymptotic behavior of positive solutions to some logarithmic Schrödinger–Poisson system

Dynamics of a Higher-Order Three-Dimensional Nonlinear System of Difference Equations

In this paper, we study the semi-cycle analysis of positive solutions and the asymptotic behavior of positive solutions of three-dimensional system of difference equations with a higher order under certain parametric conditions. Furthermore, we show the boundedness and persistence, the rate of convergence of the solutions and the global asymptotic stability of the unique equilibrium point of the proposed system under certain parametric conditions. Finally, for this system, we offer some numerical examples which support our analytical results.

Existence and Global Asymptotic Behavior of Positive Solutions for Superlinear Singular Fractional Boundary Value Problems

In this paper, we provide sufficient conditions for the existence, uniqueness and global behavior of a positive continuous solution to some nonlinear Riemann-Liouville fractional boundary value problems. The nonlinearity is allowed to be singular at the boundary. The proofs are based on perturbation techniques after reducing the considered problem to the equivalent Fredholm integral equation of the second kind. Some examples are given to illustrate our main results.

On Existence and Asymptotic Behavior of Positive Solutions for a Fractional Order Differential System Involving Riemann–Liouville Derivatives

On Existence and Asymptotic Behavior of Positive Solutions for a Fractional Order Differential System Involving Riemann–Liouville Derivatives

Nonlocal dispersal equations in domains becoming unbounded

<p style='text-indent:20px;'>This paper is concerned with a class of nonlocal dispersal equation in asymptotically cylindrical domain. Our aim is to investigate the asymptotic behavior of positive solutions in domain becoming infinite in some direction. We prove that the limiting behavior of positive solutions is only determined by the bounded part of whole domain.</p>

Existence and asymptotic behavior of nontrivial solution for Klein–Gordon–Maxwell system with steep potential well

In this paper, we consider the following nonlinear Klein–Gordon–Maxwell system with a steep potential well { − Δ u + ( λ a ( x ) + 1 ) u − μ ( 2 ω + ϕ ) ϕ u = f ( x , u ) , in R 3 , Δ ϕ = μ ( ω + ϕ ) u 2 , in R 3 , where ω > 0 is a constant, μ and λ are positive parameters, f ∈ C ( R 3 × R , R ) and the nonlinearity f satisfies the Ambrosetti–Rabinowitz condition. We use parameter-dependent compactness lemma to prove the existence of nontrivial solution for μ small and λ large enough, then explore the asymptotic behavior as μ → 0 and λ → ∞ . Moreover, we also use truncation technique to study the existence and asymptotic behavior of positive solution of Klein–Gordon–Maxwell system when f ( u ) := | u | q − 2 u where 2 < q < 4 .

Effects of boundary and spatial heterogeneity on nonlocal dispersal equations

This paper is concerned with the positive solutions of some nonlocal dispersal equations. We study the effects of boundary condition and spatial heterogeneity on the asymptotic behavior of positive solutions. By analyzing the perturbation equations, we obtain the asymptotic profiles and limiting behaviors. The results of the paper exhibit how the boundary condition and spatial heterogeneity change the patterns of nonlocal dispersal equations.

Existence and asymptotic behavior of positive solutions for a class of locally superlinear Schrödinger equation

This paper treats the existence of positive solutions of $-\Delta u + V(x) u = \lambda f(u)$ in $\mathbb{R}^N$. Here $N \geq 1$, $\lambda > 0$ is a parameter and $f(u)$ satisfies conditions only in a neighborhood of $u=0$. We shall show the existence of positive solutions with potential of trapping type or $\mathcal{G}$-symmetric potential where $\mathcal{G} \subset O(N)$. Our results extend previous results as well as we also study the asymptotic behavior of a family $(u_\lambda)_{\lambda \geq \lambda_0}$ of positive solutions as $\lambda \to \infty$.

Asymptotic behavior of positive solutions for quasilinear elliptic equations

Asymptotic behavior of positive solutions for quasilinear elliptic equations

POSITIVE SOLUTIONS FOR A NONLOCAL PROBLEM WITH CRITICAL SOBOLEV EXPONENT IN HIGHER DIMENSIONS

This paper is devoted to a nonlocal problem involving critical Sobolev exponent and negative nonlocal term. By virtue of a cut-off technique and the concentration compactness principle, we prove the existence and asymptotic behavior of positive solutions for the considered problem. In particular, our results generalize the existence results of positive solutions to higher dimensions <i>N</i> ≥ 5.

Local 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 Δ2u=upinB2∖{0} with an isolated singularity, where the punctured ball B2∖{0}⊂Rn with n>5 and p>(n+4)/(n−4). We classify isolated singularities of positive solutions and describe the asymptotic behavior of positive singular solutions without the sign assumption for −Δu.

Asymptotic behaviour of positive solutions of semilinear elliptic problems with increasing powers

We prove existence results of two solutions of the problem \[ \begin{cases} L(u)+u^{m-1}=\lambda u^{p-1} &amp; \text{in}\ \Omega,\\ u&gt;0 &amp; \text{in}\ \Omega,\\ u=0 &amp; \text{on}\ \partial \Omega, \end{cases} \] where $L(v)=-\textrm {div}(M(x)\nabla v)$ is a linear operator, $p\in (2,2^{*}]$ and $\lambda$ and $m$ sufficiently large. Then their asymptotical limit as $m\to +\infty$ is investigated showing different behaviours.

On the asymptotic behavior of positive solutions to nonlinear second-order differential equations

On the asymptotic behavior of positive solutions to nonlinear second-order differential equations

Nonlinear Schrödinger systems with mixed interactions: locally minimal energy vector solutions

This paper is concerned with asymptotic behavior of positive solutions for coupled Schrödinger equations with mixed interactions between components. We construct locally minimal energy solutions that show distinctively different limiting profile for simultaneously large attractive and repulsive couplings. The components of the solutions constructed exhibit partial synchronization and segregation.

Asymptotic Behavior of Positive Solutions to a Nonlinear Elliptic Coupled System on an Exterior Domain

Asymptotic Behavior of Positive Solutions to a Nonlinear Elliptic Coupled System on an Exterior Domain

Sharp blow up estimates and precise asymptotic behavior of singular positive solutions to fractional Hardy-Hénon equations

In this paper, we study the asymptotic behavior of positive solutions of the fractional Hardy-Hénon equation(−Δ)σu=|x|αupinB1\\{0} with an isolated singularity at the origin, where σ∈(0,1) and the punctured unit ball B1\\{0}⊂Rn with n≥2. When −2σ<α<2σ and n+αn−2σ<p<n+2σn−2σ, we give a classification of isolated singularities of positive solutions, and in particular, this implies sharp blow up estimates of singular solutions. Further, we describe the precise asymptotic behavior of solutions near the singularity. More generally, we classify isolated boundary singularities and describe the precise asymptotic behavior of singular solutions for a relevant degenerate elliptic equation with a nonlinear Neumann boundary condition. These results parallel those known for the Laplacian counterpart proved by Gidas and Spruck (1981) [21], but the methods are very different, since the ODEs analysis is a missing ingredient in the fractional case. Our proofs are based on a monotonicity formula, combined with blow up (down) arguments, Kelvin transformation and uniqueness of solutions of related degenerate equations on S+n. We also investigate isolated singularities located at infinity of fractional Hardy-Hénon equations.

Optimal integrability for some integral system of Wolff type

In the paper, we obtain the optimal integrability for positive solutions of the following integral system involving Wolff potentials: {u(x) = W?,?(vq)(x); x ?, Rn, v(x) = W?,?(up)(x), x ? Rn, where p,q &gt; 0,? &gt; 0, ? &gt; 1 and 0 &lt; ?? &lt; n. Ma, Chen and Li [Advances in Mathematics, 226(2011), 2676-2699] developed the regularity lifting method and obtained the optimal integrability for p &gt; 1, q &gt; 1. Here, based on some new observations, we overcome the difficulty there, and derive the optimal integrability for the case of p &gt; 0,q &gt; 0 and pq &gt; 1. This integrability plays a key role in estimating the asymptotic behavior of positive solutions.

Positive solutions for a class of elliptic equations

This paper is concerned with a class of semilinear elliptic equations with a potential function−Δu=λ|x|−αu−|x|σupinΩ∖{0}, where λ,σ∈R, α>0, p>1, and Ω⊂RN(N≥3) is a bounded smooth domain with 0∈Ω. We establish the existence, nonexistence and asymptotic behavior of positive solutions when the potential function |x|−α has strong singularity at the origin. When the potential function |x|−α has weak singularity at the origin, we obtain some qualitative properties of positive solutions.

Uniqueness and asymptotic behavior of positive solution of quasilinear elliptic equations with Hardy potential

In this paper, we study the uniqueness and singular behavior at the origin of positive solution to −Δpu=λup−1|x|p−|x|σf(u),x∈Ω∖{0},where N>p>1, σ>−p, 0∈Ω and Ω⊂RN(N≥3) is a domain. For the nonlinear term f(u), we suppose that limu→∞f(u)uq=a>0 and limu→0+f(u)us=b. When Ω=RN, σ≥0 and σ>N(s−p+1)−spp−1, we discuss the uniqueness of positive solution. When Ω is a bounded smooth domain, σ≥0 and σ>N(q−p+1)−pqp−1, we prove that each positive solution u(x) satisfies lim|x|→0u(x)|x|p+σq−p+1=λa+(p+σq−p+1)p−1(1+p+σq−p+1)p−1a−N−1a1q−p+1,and the equation with the boundary condition u=ϕ≥0 on ∂Ω has a unique positive solution. When Ω=BR(0) and σ≥0, the uniqueness of positive solution to the corresponding Dirichlet problem is also established.