Elliptic Equations In Unbounded Domains Research Articles

Uniqueness of solution of boundary value problems for the biharmonic equation

AbstractIt is known that in order for a solution of a boundary value problems for elliptic equations in unbounded domains to be unique, it is necessary to impose additional constraints on the behavior, of solutions at infinity. Such constrains may be qualitatively different.In [1] it was noticed that the assumption that the solutions should be positive for sufficiently large arguments may also serve as a condition that guarantees the uniqueness of solutions. The present report explains an investigation of the conditions on boundary value problems whose fulfillment guarantees the validity of such uniqueness theorems for the biharmonic equation. For harmonic functions that satisfy the Dirichlet or Neumann boundary conditions, there are no such uniqueness theorems. But they can exist for simplest nonlocal boundary value problems for harmonic functions.

On the existence of a positive solution of semilinear elliptic equations in unbounded domains

We prove here the existence of a positive solution, under general conditions, for semilinear elliptic equations in unbounded domains with a variational structure. The solutions we build cannot be obtained in general by minimization problems. And even if Palais-Smale condition is violated, precise estimates on the losses of compactness are obtained by the concentration-compactness method which enables us to apply the theory of critical points at infinity.

Least energy solutions for elliptic equations in unbounded domains

In this paper we study the existence of least energy solutions to subcritical semilinear elliptic equations of the formwhere Ω is an unbounded domain inRNandfis aC1function, with appropriate superlinear growth. We state general conditions on the domain Ω so that the associated functional has a nontrivial critical point, thus yielding a solution to the equation. Asymptotic results for domains stretched in one direction are also provided.

Monotone methods for semilinear elliptic equations in unbounded domains

Monotone methods in conjuction with upper and lower solutions have proved to be extremely powerful tools in the study of nonlinear elliptic boundary value problems in bounded domains. In this paper the theory is extended to unbounded (not necessarily exterior) domains and general linear boundary conditions.

Limitations on the rate of decrease of solutions of elliptic equations in unbounded domains

Limitations on the rate of decrease of solutions of elliptic equations in unbounded domains

Behavior of solutions of the Dirichlet problem for quasilinear divergent higher-order elliptic equations in unbounded domains

In this article we determine the energy apriori estimates of solutions of the Dirichlet problem in unbounded domains with noncompact boundaries. These estimates depend on the geometry of the boundary, and in the case of second-order equations, for a wide class of domains they coincide with those previously known [I, 2]. The method of obtaining these estimates is a new one, conceptually similar to the method used in [3, 4] to obtai n interior estimates of generalized solutions. On the basis of established energy estimates we prove alternative theorems of the type of the Phragmen--Lindel0f Theorem on the behavior of solutions at infinity. For linear equations, the first such theorems in terms of an interior diameter of the domain are given in [5]; in the same terms, for solutions of quasilinear divergent higher order elliptic equations, such a result is announced in [6]. We note further that for solutions of a polyharmonic equation in the case of some model domains, energy apriori estimates of the type of Saint-Venant's principle were established in [7].

Estimate of solution of quasilinear elliptic equation in unbounded domain

Estimate of solution of quasilinear elliptic equation in unbounded domain

Minimum action solutions of non-linear elliptic equations in unbounded domains containing a plane

SynopsisLet d ≧ 1 be an integer and ω ⊂ℝd a smooth bounded domain and consider the elliptic equation − Δu = g(u) on Ω = ℝ2 × ω. We prove that under (almost) necessary and sufficient conditions on the continuous function g: ℝm→ ℝm the above equation has a minimum-action solution.

Remarks on Certain Non-Linear Elliptic Equations in Unbounded Domains

On etudie l'equation superlineaire (Δu) (x)−u(x)=f(x)+|u(x)| m+ e, x∈R n , ou m∈N et 0<e≤1 dans le cadre des espaces de Sobolev W p l (R n ) et des espaces de Besov B p,q s (R n )

Positive solutions of linear and quasilinear elliptic equations in unbounded domains

Soit Ω un domaine exterieur dans R N , N≥2, a frontiere lisse Γ=∂Ω et soeint #7B-D et B un operateur elliptique et un operateur frontiere. On etudie l'existence de solutions positives aux problemes aux valeurs limites suivants: −#7B-Du+c(x)u=λm(x)u dans Ω, Bu=0 sur Γ et −#7B-Du+c(x)u=λm(x)uγ dans Ω, Bu=0 sur Γ, ou c(x) et m(x) sont des fonctions donnees, λ est un parametre reel et γ est une constante non nulle avec γ¬=1

Asymptotics of the spectrum of variational problems on solutions of elliptic equations in unbounded domains

Asymptotics of the spectrum of variational problems on solutions of elliptic equations in unbounded domains