Infinitely many solutions of a symmetric semilinear elliptic equation on an unbounded domain

We study a semilinear elliptic equation of the form

Dirichlet problem for a divergence form elliptic equation with unbounded coefficients in an unbounded domain

We prove existence and uniqueness of the solution of the Dirichlet problem for a class of elliptic equations in divergence form with discontinuous and unbounded coefficients in unbounded domains. Entrata in Redazione il 22 aprile 1999.     

Behavior of solutions of quasilinear elliptic inequalities in an unbounded domain

We consider the solutions of the inequalityLu≤?(¦gradu¦), whereL is a uniformly elliptic homogeneous operator and ? is a function increasing faster than any linear function but not faster thanξ lnξ, in the unbounded domain $$\left\{ {x \in \mathbb{R}^n |\sum\limits_{i = 2}^n {x_i^2< (\psi (x_1 ))^2 ,} {\text{ }} - \infty< x_1< \infty } \right\},$$ , whereψ is a bounded function with bounded derivative. We estimate the growth of the solutions in terms of $\int_0^{x_1 } {(1/\psi (r))dr}$ . For the special case in which?(ξ)=aξ lnξ+C, the solutionsu(x ₁,x ₂,...,x _n ) grow as $\left( {\int_0^{x_1 } {(1/\psi (r))dr} } \right)^N$ , whereN is any given number anda=a(N).     

Global attractor and nonhomogeneous equilibria for a nonlocal evolution equation in an unbounded domain

In this work we consider the nonlocal evolution equation which arises in models of phase separation. We prove the existence of a compact global attractor in some weighted spaces and the existence of a distinguished nonhomogeneous equilibrium: the ‘critical droplet.’     

The attractor for a nonlinear reaction‐diffusion system in an unbounded domain

In this paper the quasi‐linear second‐order parabolic systems of reaction‐diffusion type in an unbounded domain are considered. Our aim is to study the long‐time behavior of parabolic systems for which the nonlinearity depends explicitly on the gradient of the unknown functions. To this end we give a systematic study of given parabolic systems and their attractors in weighted Sobolev spaces. Dependence of the Hausdorff dimension of attractors on the weight of the Sobolev spaces is considered. © 2001 John Wiley & Sons, Inc.     

On the Neumann problem for higher-order divergent elliptic equations in an unbounded domain,close to a cylinder

The Neumann problem for a higher-order divergent elliptic equation, defined in an unbounded domain, close to a cylinder, is investigated. It is proved that each solution, having a slowly increasing energy integral, tends at infinity to a certain polynomial and, in the case of an exponential decrease of the righthand side of the equation, the convergence rate is also exponential. Existence and uniqueness are obtained in classes of functions with bounded or unbounded energy integral. Formulas, expressing the coefficients of the limit polynomial in terms of the right-hand side of the equation and of the Dirichlet data at the base of an unbounded domain, close to a semiinfinite cylinder, are derived.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 16, pp. 191–217, 1992.     

Numerical approximation of bounded solutions for semilinear elliptic equations in an unbounded cylindrical domain

We consider the approximation of bounded solutions for a class of semilinear elliptic equations in an unbounded cylindrical domain. On certain conditions, the existence of approximate solutions for the semidiscretization of the problem is proved. Error estimates, applications, and a numerical example are also given. © 1993 John Wiley & Sons, Inc.     

Existence of a global attractor for the parabolic equation with nonlinear Laplacian principal part in an unbounded domain

In this paper, we study the long-time behavior of solutions for the parabolic equation with non-linear Laplacian principal part in Rn. We prove the existence of a global (L₂(Rn), L_∞(Rn))-attractor when n?p and the existence of a global (L₂(Rn), )-attractor when n>p.     

On an elliptic equation of Kirchhoff-type with a potential asymptotically linear at infinity

This paper examines the existence of positive solutions for a boundary value problem of Kirchhoff-type involving a positive potential function which is asymptotically linear at infinity.     

Radial solution of asymptotically linear elliptic equation with mixed boundary value in annular domain

In this paper, we study nonlinear elliptic equation with mixed boundary value condition in annular domain. It is assumed that the nonlinearity is asymptotically linear and depends on the derivative term. Some results on the existence of solution are established by nonlinear analysis methods.     

Existence and multiplicity results for singular quasilinear elliptic equation on unbounded domain

In this paper, we are interested in the existence and multiplicity results of solutions for the singular quasilinear elliptic problem with concave–convex nonlinearities (0.1) where is an unbounded exterior domain with smooth boundary ?Ω, 1 < p < N,0 ≤ a < (N ? p) ∕ p,λ > 0,1 < s < p < r < q = pN ∕ (N ? pd),d = a + 1 ? b,a ≤ b < a + 1. By the variational methods, we prove that problem 0.1 admits a sequence of solutions u_k under the appropriate assumptions on the weight functions H(x) and H(x). For the critical case, s = q,h(x) = | x | ^{? bq}, we obtain that problem 0.1 has at least a nonnegative solution with p < r < q and a sequence of solutions u_k with 1 < r < p < q and J(u_k) → 0 as k → ∞ , where J(u) is the energy functional associated to problem 0.1 . Copyright © 2013 John Wiley & Sons, Ltd.