Elliptic Problems In Unbounded Domains Research Articles

Existence of Pulses for Monotone Reaction-Diffusion Systems

Existence of pulse solutions, that is, positive stationary solutions with zero limit at infinity is studied for monotone reaction-diffusion systems in the bistable case. It is shown that such solutions exist if and only if the speed of the travelling wave described by the same system is positive. The proof is based on the Leray--Schauder method using topological degree for elliptic problems in unbounded domains and a priori estimates of solutions in some appropriate weighted spaces.

Existence of pulses for a reaction-diffusion system of blood coagulation

The paper is devoted to the investigation of a reaction-diffusion system of equations describing the process of blood coagulation. Existence of pulse solutions, that is, positive stationary solutions with zero limit at infinity is studied. It is shown that such solutions exist if and only if the speed of the travelling wave described by the same system is positive. The proof is based on the Leray-Schauder method using topological degree for elliptic problems in unbounded domains and a priori estimates of solutions in some appropriate weighted spaces.

Positive solutions for autonomous and non-autonomous nonlinear critical elliptic problems in unbounded domains

The paper concerns with positive solutions of problems of the type $$-\Delta u+a(x)\, u=u^{p-1}+\varepsilon u^{2^*-1}$$ in $$\Omega \subseteq \mathbb {R}^N$$, $$N\ge 3$$, $$2^*={2N\over N-2}$$, $$2 0$$. First, some existence results of ground state solutions are proved. Then the case $$a(x)\ge a_\infty $$ is considered, with $$a(x)\not \equiv a_\infty $$ or $$\Omega \ne \mathbb {R}^N$$. In such a case, no ground state solution exists and the existence of a bound state solution is proved, for small $${\varepsilon }$$.

On Some Elliptic Problems in Unbounded Domains

The author presents a method allowing to obtain existence of a solution for some elliptic problems set in unbounded domains, and shows exponential rate of convergence of the approximate solution toward the solution.

Inverted finite elements for degenerate and radial elliptic problems in unbounded domains

This paper is the second in a series devoted to the development of inverted finite element method for elliptic problems in unbounded domains. Here, focus is on degenerate elliptic equations having singular or/and unbounded coefficients. A specific emphasis is being placed on radial solutions of second order elliptic equations. After giving an exposition of the fundamentals of the method and proving its convergence, we display some computational tests which demonstrate its efficiency.

Multiplicity results for some perturbed elliptic problems in unbounded domains with non-homogeneous boundary conditions

We study the following nonlinear elliptic problem {−Δu+mu=g(x,u)+f(x)in RN∖B¯R,u=hon ∂BR,u→0as ∣x∣→+∞, with N≥3, m∈R+∗, under suitable conditions on the radial functions g, f and h. Multiplicity results are proved in the perturbed case f≢0 and h≢0 by using Bolle’s perturbation method and suitable growth estimates on min–max critical levels.

Existence of a ground state solution for a singular elliptic problem in unbounded domain and dimension 2

In this paper we prove the existence of ground state solutions for a class of singular elliptic problems in unbounded domains of R2 with Dirichlet boundary conditions and nonlinearity with subcritical growth.

Numerical Approximation of Second-Order Elliptic Problems in Unbounded Domains

This paper deals with the numerical resolution of elliptic problems in unbounded domains using inverted finite elements. In opposition to conventional approaches which are based on the truncation of the domain, the suggested method keeps the domain unbounded and is based on a description of the asymptotic behavior in an appropriate functional framework. The method and its mathematical properties are presented first, and some computational examples are carried out. The obtained numerical results demonstrate the efficiency of the method.

Existence of Reaction-Diffusion Waves with Nonlinear Boundary Conditions

The paper is devoted to a reaction-diffusion equation in an infinite two-dimensional strip with nonlinear boundary conditions. The existence of travelling waves is proved in the bistable case by the Leray-Schauder method. It is based on a topological degree for elliptic problems in unbounded domains and on a priori estimates of solutions.

Numerical solution of bi-periodic elliptic problems in unbounded domains

Numerical solution of bi-periodic elliptic problems in unbounded domains

On Some Nonhomogeneous Elliptic Problems in Unbounded Domains

Abstract In this paper we consider the problem where Ω = ℝN ∖ ω̅, ω being a nonempty open bounded domain of ℝN having smooth boundary We prove that, if u0 and f are nonnegative and satisfy suitable conditions, there exist at least two positive solutions. Moreover, the existence of a solution is shown to hold even when no condition on the sign of u0 and f is assumed.

Elliptic problems with a parameter in unbounded domains

Linear elliptic boundary-value problems with a parameter are studied. The Agranovich-Vishik method and specially introduced function spaces allow us to consider mixed-order problems in unbounded domains. We obtain a priori estimates and unique solvability for large values of the parameter. These results are used to study analytic semi-groups and Fredholm property of general elliptic problems in unbounded domains.

Some Nonlinear Elliptic Problems in Unbounded Domains

In this paper the results of some investigations concerning nonlinear elliptic problems in unbounded domains are summarized and the main difficulties and ideas related to these researches are described. The model problem $$ (P)\quad \left\{ \begin{array}{*{20}rc} -\Delta u + a(x)u = |u|^{p - 2} u& {\text{in }}\Omega ,\\ u \in H_0^1 (\Omega ), \end{array} \right. $$ where $$\Omega \subseteq \mathbb{R}^N $$ , N ≥ 3, is an unbounded smooth domain, a(x) is a smooth real function defined on Ω, such that $$a(x)\xrightarrow[{|x| \to + \infty }]{}a_\infty > 0,\; p \in \left( {2,\frac{{2N}}{{N - 2}}} \right)$$ , is considered and existence and multiplicity results are given under various assumptions on Ω.

Multiple Positive Solutions For Nonautonomous Quasicritical Elliptic Problems in Unbounded Domains

Abstract The problem with boundary Dirichlet zero data is considered in an exterior domain Ω ⊂ ℝN. Assuming that, as ε →0, aε concentrates and blows up at a point of Ω, namely the existence of at least 2 distinct positive solutions is proved, if is suitably small. Furthermore, if aε(x) has a suitable behaviour at infinity, the existence of another positive solution is shown.

Multiplicity of positive solutions for semilinear elliptic problems in unbounded domains

In this paper, we study the effect of domain shape on the multiplicity of positive solutions for the semilinear elliptic equations. We prove a Palais–Smale condition in unbounded domains and assert that the semilinear elliptic equation in unbounded domains has multiple positive solutions.

Positive solutions of semilinear elliptic problems in unbounded domains with unbounded boundary

This paper is concerned with the existence and multiplicity of positive solutions of the equation −Δu+u=up−1, 2<p<2∗=2NN−2, with Dirichlet zero data, in an unbounded smooth domain Ω⊂RN having unbounded boundary. Under the assumptions:(h1)∃τ1,τ2,…,τk∈R+∖{0}, 1⩽k⩽N−2, such that(x1,x2,…,xN)∈Ω⟺(x1,…,xi−1,xi+τi,…,xN)∈Ω,∀i=1,2,…,k,(h2)∃R∈R+∖{0} such that RN∖Ω⊂{(x1,x2,…,xN)∈RN:∑j=k+1Nxj2⩽R2} the existence of at least k+1 solutions is proved.

Semilinear elliptic problems in unbounded domains

We investigate the existence of positive solutions and their continuous dependence on functional parameters for a semilinear Dirichlet problem. We discuss the case when the domain is unbounded and the nonlinearity is smooth and convex on a certain interv

Existence of positive solutions of some nonlinear elliptic problems in unbounded domains

In this paper, we study the existence of positive solutions of some nonlinear elliptic problems in unbounded domains. The existence is affected by the properties of the geometry and the topology of the domain.

Existence of reaction‐diffusion‐convection waves in unbounded strips

Existence of reaction‐diffusion‐convection waves in unbounded strips is proved in the case of small Rayleigh numbers. In the bistable case the wave is unique, in the monostable case they exist for all speeds greater than the minimal one. The proof uses the implicit function theorem. Its application is based on the Fredholm property, index, and solvability conditions for elliptic problems in unbounded domains.

Inverted finite elements: a new method for solving elliptic problems in unbounded domains

In this paper, we propose a new numerical method for solving elliptic equations in unbounded regions of . The method is based on the mapping of a part of the domain into a bounded region. An appropriate family of weighted spaces is used for describing the growth or the decay of functions at large distances. After exposing the main ideas of the method, we analyse carefully its convergence. Some 3D computational results are displayed to demonstrate its efficiency and its high performance.