Existence and asymptotic behavior of solutions for a class of semilinear subcritical elliptic systems

2021 ◽  
pp. 1-20
Author(s):  
Suellen Cristina Q. Arruda ◽  
Giovany M. Figueiredo ◽  
Rubia G. Nascimento
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Behaviour ◽  
Variational Methods ◽  
Positive Solution ◽  
Existence Of Solutions ◽  
Homogeneous Function ◽  
Elliptic Systems ◽  
Iteration Scheme ◽  
Asymptotic Behavior Of Solutions ◽  
Subcritical Growth

In this paper we study the asymptotic behaviour of a family of elliptic systems, as far as the existence of solutions is concerned. We give a special attention to the asymptotic behaviour of W and V as ε goes to zero in the system − ε 2 Δ u + W ( x ) u = Q u ( u , v ) in  R N , − ε 2 Δ v + V ( x ) v = Q v ( u , v ) in  R N , u , v ∈ H 1 ( R N ) , u ( x ) , v ( x ) > 0 for each  x ∈ R N , where ε > 0, W and V are positive potentials of C 2 class and Q is a p-homogeneous function with subcritical growth. We establish the existence of a positive solution by considering two classes of potentials W and V. Our arguments are based on penalization techniques, variational methods and the Moser iteration scheme.

scholarly journals SYMMETRY AND EXISTENCE OF SOLUTIONS OF SEMI-LINEAR ELLIPTIC SYSTEMS IN HYPERBOLIC SPACE

2014 ◽  
Vol 57 (3) ◽  
pp. 519-541
Author(s):  
HAIYANG HE
Keyword(s):  
Positive Solution ◽  
Hyperbolic Space ◽  
Existence Of Solutions ◽  
Elliptic Systems ◽  
Ground State Solution ◽  
Decay Estimates ◽  
Symmetry Properties ◽  
Formula Group ◽  
Solution Pair ◽  
Image Position

Abstract(0.1) \begin{equation}\label{eq:0.1} \left\{ \begin{array}{ll} \displaystyle -\Delta_{\mathbb{H}^{N}}u=|v|^{p-1}v x, \\ \displaystyle -\Delta_{\mathbb{H}^{N}}v=|u|^{q-1}u, \\ \end{array} \right. \end{equation} in the whole Hyperbolic space ℍN. We establish decay estimates and symmetry properties of positive solutions. Unlike the corresponding problem in Euclidean space ℝN, we prove that there is a positive solution pair (u, v) ∈ H1(ℍN) × H1(ℍN) of problem (0.1), moreover a ground state solution is obtained. Furthermore, we also prove that the above problem has a radial positive solution.

scholarly journals Asymptotic Properties of Solutions to Second-Order Difference Equations of Volterra Type

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Janusz Migda ◽  
Małgorzata Migda ◽  
Magdalena Nockowska-Rosiak
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equations ◽  
Existence Of Solutions ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Type Equation ◽  
Asymptotic Behavior Of Solutions ◽  
Volterra Type ◽  
Order Difference ◽  
Prescribed Asymptotic Behavior

We consider the discrete Volterra type equation of the form Δ(rnΔxn)=bn+∑k=1nK(n,k)f(xk). We present sufficient conditions for the existence of solutions with prescribed asymptotic behavior. Moreover, we study the asymptotic behavior of solutions. We use o(ns), for given nonpositive real s, as a measure of approximation.

Well-Posedness and Asymptotic Behavior of a Nonclassical Nonautonomous Diffusion Equation with Delay

2015 ◽  
Vol 25 (14) ◽  
pp. 1540021 ◽  
Author(s):  
Tomás Caraballo ◽  
Antonio M. Márquez-Durán ◽  
Felipe Rivero
Keyword(s):  
Fixed Point ◽  
Asymptotic Behavior ◽  
Diffusion Equation ◽  
Existence Of Solutions ◽  
Local Solution ◽  
Asymptotic Behavior Of Solutions ◽  
Well Posedness ◽  
Pullback Attractors ◽  
Nonautonomous Dynamical Systems ◽  
Equation With Delay

In this paper, a nonclassical nonautonomous diffusion equation with delay is analyzed. First, the well-posedness and the existence of a local solution is proved by using a fixed point theorem. Then, the existence of solutions defined globally in future is ensured. The asymptotic behavior of solutions is analyzed within the framework of pullback attractors as it has revealed a powerful theory to describe the dynamics of nonautonomous dynamical systems. One difficulty in the case of delays concerns the phase space that one needs to construct the evolution process. This yields the necessity of using a version of the Ascoli–Arzelà theorem to prove the compactness.

A general monotone scheme for elliptic systems with applications to ecological models

1986 ◽  
Vol 102 (3-4) ◽  
pp. 315-325 ◽  
Author(s):  
Philip Korman ◽  
Anthony W. Leung
Keyword(s):  
Existence Of Solutions ◽  
Elliptic Systems ◽  
Species Interaction ◽  
Iteration Scheme ◽  
Steady States ◽  
Ecological Models ◽  
Monotone Scheme ◽  
Weakly Coupled ◽  
Image Position

SynopsisWe consider weakly-coupled elliptic systems of the typewith each fi being either an increasing or a decreasing function of each uj. Assuming the existence of coupled super- and subsolutions, we prove the existence of solutions, and provide a constructive iteration scheme to approximate the solutions. We then apply our results to study the steady-states of two-species interaction in the Volterra–Lotka model with diffusion.

QUALITATIVE ANALYSIS OF SEMILINEAR CATTANEO EQUATIONS

1998 ◽  
Vol 08 (03) ◽  
pp. 507-519 ◽  
Author(s):  
T. HILLEN
Keyword(s):  
Wave Propagation ◽  
Asymptotic Behavior ◽  
Transport Theory ◽  
Existence Of Solutions ◽  
Stationary Solutions ◽  
Asymptotic Behavior Of Solutions ◽  
Limit Set ◽  
Correlated Random Walk ◽  
Finite Speed ◽  
Global Existence Of Solutions

The linear Cattaneo equation appears in heat transport theory to describe heat wave propagation with finite speed. It can also be seen as a generalization of a correlated random walk. If the system admits nonconservative forces (or reactions), then a nonlinear Cattaneo system is obtained. Here we consider asymptotic behavior of solutions of the nonlinear Cattaneo system. Following Brayton and Miranker we define a Lyapunov function to show global existence of solutions and to show that each ω-limit set is contained in the set of all stationary solutions.

scholarly journals Existence of positive solutions for a class of the p-Laplace equations

1994 ◽  
Vol 36 (2) ◽  
pp. 249-264 ◽  
Author(s):  
Yin Xi Huang
Keyword(s):  
Asymptotic Behavior ◽  
Positive Solution ◽  
Existence Result ◽  
Existence Of Solutions ◽  
Unique Positive Solution ◽  
Bounded Smooth Domain ◽  
Laplace Equations ◽  
Existence Of Positive Solutions ◽  
Bounded Smooth ◽  
Image Position

AbstractWe are concerned with the existence of solutions ofwhere Δp is the p-Laplacian, p ∈ (1, ∞), and Ω is a bounded smooth domain in ℝn.For h(x) ≡ 0 and f(x, u) satisfying proper asymptotic spectral conditions, existence of a unique positive solution is obtained by invoking the sub-supersolution technique and the spectral method. For h(x) ≢ 0, with assumptions on asymptotic behavior of f(x, u) as u → ±∞, an existence result is also proved.

EXISTENCE OF POSITIVE SOLUTION FOR INDEFINITE KIRCHHOFF EQUATION IN EXTERIOR DOMAINS WITH SUBCRITICAL OR CRITICAL GROWTH

2016 ◽  
Vol 103 (3) ◽  
pp. 329-340 ◽  
Author(s):  
G. M. FIGUEIREDO ◽  
D. C. DE MORAIS FILHO
Keyword(s):  
Variational Methods ◽  
Boundary Value Problems ◽  
Positive Solution ◽  
Exterior Domain ◽  
Critical Case ◽  
Existence Of Solutions ◽  
Nonlocal Boundary ◽  
Exterior Domains ◽  
Subcritical Case ◽  
Image Position

Using variational methods and depending on a parameter $\unicode[STIX]{x1D706}$ we prove the existence of solutions for the following class of nonlocal boundary value problems of Kirchhoff type defined on an exterior domain $\unicode[STIX]{x1D6FA}\subset \mathbb{R}^{3}$: $$\begin{eqnarray}\left\{\begin{array}{@{}ll@{}}M(\Vert u\Vert ^{2})[-\unicode[STIX]{x1D6E5}u+u]=\unicode[STIX]{x1D706}a(x)g(u)+\unicode[STIX]{x1D6FE}|u|^{4}u\quad & \text{in }\unicode[STIX]{x1D6FA},\\ u=0\quad & \text{on }\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA},\end{array}\right.\end{eqnarray}$$ for the subcritical case ($\unicode[STIX]{x1D6FE}=0$) and also for the critical case ($\unicode[STIX]{x1D6FE}=1$).

scholarly journals Asymptotic Behavior of Solutions to a Vector Integral Equation with Deviating Arguments

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Cristóbal González ◽  
Antonio Jiménez-Melado
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Asymptotic Behavior ◽  
Existence Of Solutions ◽  
Sufficient Conditions ◽  
Asymptotic Behavior Of Solutions ◽  
Abstract Differential Equation ◽  
Deviating Arguments ◽  
Vector Integral ◽  
Similar Equation

In this paper, we propose the study of an integral equation, with deviating arguments, of the typey(t)=ω(t)-∫0∞‍f(t,s,y(γ1(s)),…,y(γN(s)))ds,t≥0,in the context of Banach spaces, with the intention of giving sufficient conditions that ensure the existence of solutions with the same asymptotic behavior at∞asω(t). A similar equation, but requiring a little less restrictive hypotheses, isy(t)=ω(t)-∫0∞‍q(t,s)F(s,y(γ1(s)),…,y(γN(s)))ds,t≥0.In the case ofq(t,s)=(t-s)+, its solutions with asymptotic behavior given byω(t)yield solutions of the second order nonlinear abstract differential equationy''(t)-ω''(t)+F(t,y(γ1(t)),…,y(γN(t)))=0,with the same asymptotic behavior at∞asω(t).

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