Elliptic Equations In Unbounded Domains Research Articles

Existence, Regularity, and Uniqueness of Solutions to Some Noncoercive Nonlinear Elliptic Equations in Unbounded Domains

Existence, Regularity, and Uniqueness of Solutions to Some Noncoercive Nonlinear Elliptic Equations in Unbounded Domains

On the Oscillation Properties of Elliptic Equations in Unbounded Domains

On the Oscillation Properties of Elliptic Equations in Unbounded Domains

Monotonicity of positive solutions to quasilinear elliptic equations in half-spaces with a changing-sign nonlinearity

In this paper we prove the monotonicity of positive solutions to \( -\Delta _p u = f(u) \) in half-spaces under zero Dirichlet boundary conditions, for \((2N+2)/(N+2)< p < 2\) and for a general class of regular changing-sign nonlinearities f. The techniques used in the proof of the main result are based on a fine use of comparison and maximum principles and on an adaptation of the celebrated moving plane method to quasilinear elliptic equations in unbounded domains.

Classification and Liouville-type theorems for semilinear elliptic equations in unbounded domains

We classify stable and finite Morse index solutions to general semilinear elliptic equations posed in Euclidean space of dimension at most 10, or in some unbounded domains.

Symmetry properties of stable solutions of semilinear elliptic equations in unbounded domains

We consider stable solutions of a semilinear elliptic equation with homogeneous Neumann boundary conditions. A classical result of Casten, Holland and Matano states that all stable solutions are constant in convex bounded domains. In this paper, we examine whether this result extends to unbounded convex domains. We give a positive answer for stable non-degenerate solutions, and for stable solutions if the domain \(\Omega \) further satisfies \(\Omega \cap \{\vert x\vert \le R\}= O(R^2)\), when \(R\rightarrow +\infty \). If the domain is a straight cylinder, an additional natural assumption is needed. These results can be seen as an extension to more general domains of some results on De Giorgi’s conjecture. As an application, we establish asymptotic symmetries for stable solutions when the domain satisfies a geometric property asymptotically.

Mixed boundary value problems for non-divergence type elliptic equations in unbounded domains

Mixed boundary value problems for non-divergence type elliptic equations in unbounded domains

Best possible estimates of weak solutions of boundary value problems for quasi-linear elliptic equations in unbounded domains

Abstract We investigate the behaviour of weak solutions of boundary value problems for quasi-linear elliptic divergence second order equations in unbounded domains. We show the boundedness of weak solutions to our problem. Using barrier function and applying the comparison principle, we find the exact exponent of weak solutions decreasing rate near the infinity.

Shear Flows of an Ideal Fluid and Elliptic Equations in Unbounded Domains

We prove that, in a two‐dimensional strip, a steady flow of an ideal incompressible fluid with no stationary point and tangential boundary conditions is a shear flow. The same conclusion holds for a bounded steady flow in a half‐plane. The proofs are based on the study of the geometric properties of the streamlines of the flow and on one‐dimensional symmetry results for solutions of some semilinear elliptic equations. Some related rigidity results of independent interest are also shown in n‐dimensional slabs in any dimension n.© 2016 Wiley Periodicals, Inc.

ABP and global Hölder estimates for fully nonlinear elliptic equations in unbounded domains

We prove global Hölder estimates for solutions of fully nonlinear elliptic or degenerate elliptic equations in unbounded domains under geometric conditions à la Cabré.

Multiple positive solutions for a class of concave–convex elliptic problems in ℝNinvolving sign-changing weight, II

In this paper, we study the following concave–convex elliptic problems: [Formula: see text] where N ≥ 3, 1 &lt; q &lt; 2 &lt; p &lt; 2* = 2N/(N - 2), λ &gt; 0 and μ &lt; 0 are two parameters. By using several variational methods and a perturbation argument, we obtain three positive solutions to this problem under the predefined conditions of fλ(x) and gμ(x), which simultaneously extends the result of [T. Hsu, Multiple positive solutions for a class of concave–convex semilinear elliptic equations in unbounded domains with sign-changing weights, Bound. Value Probl. 2010 (2010), Article ID 856932, 18pp.; T. Wu, Multiple positive solutions for a class of concave–convex elliptic problems in ℝNinvolving sign-changing weight, J. Funct. Anal. 258 (2010) 99–131]. We also study the concentration behavior of these three solutions both for λ → 0 and μ → -∞.

On some $L^{p}$-estimates for solutions of elliptic equations in unbounded domains

In this review article we present an overview on some a priori estimates in $L^p$, $p>1$, recently obtained in the framework of the study of a certain kind of Dirichlet problem in unbounded domains. More precisely, we consider a linear uniformly elliptic second order differential operator in divergence form with bounded leading coeffcients and with lower order terms coefficients belonging to certain Morrey type spaces. Under suitable assumptions on the data, we first show two $L^p$-bounds, $p>2$, for the solution of the associated Dirichlet problem, obtained in correspondence with two different sign assumptions. Then, putting together the above mentioned bounds and using a duality argument, we extend the estimate also to the case $1<p<2$, for each sign assumption, and for a data in $L^p\cap L^2$.

The Dirichlet problem for elliptic equations in weighted Sobolev spaces on unbounded domains of the plane

Abstract This paper deals with the Dirichlet problem for second order linear elliptic equations in unbounded domains of the plane in weighted Sobolev spaces. We prove an a priori bound and an existence and uniqueness result.

Weak solutions for a class of semilinear elliptic equations in unbounded domains of ℝ3

We study the existence of a weak solution for a semilinear elliptic Dirichlet boundary-value problem in a suitable weighted Sobolev space, where Ω = ℝ3\\K is an unbounded domain, and where K is a closure of some bounded domain in ℝ3.

Asymptotic behavior of solutions of semilinear elliptic equations in unbounded domains: Two approaches

In this paper, we study the asymptotic behavior as x 1 → + ∞ of solutions of semilinear elliptic equations in quarter- or half-spaces, for which the value at x 1 = 0 is given. We prove the uniqueness and characterize the one-dimensional or constant profile of the solutions at infinity. To do so, we use two different approaches. The first one is a pure PDE approach and it is based on the maximum principle, the sliding method and some new Liouville type results for elliptic equations in the half-space or in the whole space R N . The second one is based on the theory of dynamical systems.

Existence and multiplicity of positive solutions of semilinear elliptic equations in unbounded domains

We investigate the existence and the multiplicity of positive solutions for the semilinear elliptic equation − Δ u + u = Q ( x ) | u | p − 2 u in exterior domain which is very close to R N . The potential Q ( x ) tends to positive constant at infinity and may change sign.

Existence of 2–Nodal Solutions for Semilinear Elliptic Equations in Unbounded Domains

Abstract In this paper, we study the effect of domain shape on the existence of 2-nodal solutions for a semilinear elliptic equation involving non-odd nonlinearities.

Multiple Positive Solutions for a Class of Concave-Convex Semilinear Elliptic Equations in Unbounded Domains with Sign-Changing Weights

We study the existence and multiplicity of positive solutions for the following Dirichlet equations: in , on , where , ( if ; if ), is a smooth unbounded domain in , , and satisfy suitable conditions, and maybe change sign in .

Behaviour at infinity of solutions of pseudodifferential elliptic equations in unbounded domains

Upper bounds are obtained for solutions of the Dirichlet problem for pseudodifferential elliptic equations where the right-hand side has compact support. In domains with non-compact boundary they characterise the behaviour of solutions at infinity in its dependence on the geometric properties of the domain. For unbounded domains where the boundary has irregular behaviour, it is shown that these bounds may be more efficient than the bounds that are already known for second-order elliptic equations. For second-order elliptic equations in a broad class of domains of revolution these bounds are shown to be sharp. Bibliography: 17 titles.

Existence of weak solutions for a class of nonuniformly nonlinear elliptic equations in unbounded domains

The goal of this paper is to study the existence of non-trivial weak solutions for the nonuniformly nonlinear elliptic equation − div ( h ( x ) ∇ u ) + q ( x ) u = f ( x , u ) in an unbounded domain Ω ⊂ R N ( N ≥ 3 ) , where h ( x ) ∈ L loc 1 ( Ω ) . The solutions will be obtained in a subspace of the Sobolev space H 0 1 ( Ω ) and the proofs rely essentially on a variation of the Mountain pass theorem in [D.M. Duc, Nonlinear singular elliptic equations, J. London. Math. Soc. 40 (2) (1989) 420–440].

The Dirichlet Problem for elliptic equations in unbounded domains of the plane

In this paper we prove a uniqueness and existence theorem for the Dirichlet problem in W2,p for second order linear elliptic equations in unbounded domains of the plane. Here the leading coefficients are locally of class VMO and satisfy a suitable condition at infinity.