Volume 5, Issue 4

We consider a system of weakly coupled singularly perturbed semilinear elliptic equations. First, we obtain a Lipschitz regularity result for the associated ground energy function Σ as well as representation formulas for the left and the right derivatives. Then, we show that the concentration points of the solutions locate close to the critical points of Σ in the sense of subdifferential calculus.

The averaging method has been used to study random or non-autonomous systems on a fast time scale. We apply this method to a random abstract evolution equation on a fast time scale whose long time behavior can be characterized by a random attractor or a random inertial manifold. The main purpose is to show that the long-time behavior of such a system can be described by a deterministic evolution equation with averaged coefficients. Our first result provides an averaging result on finite time intervals which we use to show that under a dissipativity assumption the attractors of the fast time scale systems are upper semicontinuous when the scaling parameter goes to zero. Our main result deals with a global averaging procedure. Under some spectral gap condition we show that inertial manifolds of the fast time scale system tend to an inertial manifold of the averaged system when the scaling parameter goes to zero. These general results can be applied to semilinear parabolic differential equations containing a scaled ergodic noise on a fast time scale which includes scaled almost periodic motions.

The purpose of this paper is to investigate the dynamics of a class of triangular parabolic systems given on bounded domains of arbitrary dimension. In particular, the existence of global attractors and the persistence property will be established.

We consider a class of semilinear elliptic equations of the form −Δu(x,y) + a(εx)Wʹ(u(x,y)) = 0, (x,y) ∊ ℝ 2 (0.1) where ε > 0, a : ℝ → ℝ is an almost periodic, positive function and W : ℝ → ℝ is modeled on the classical two well Ginzburg-Landau potential W(s) = (s 2 - 1) 2 . We show via variational methods that if ε is sufficiently small and a is not constant then (0.1) admits infinitely many two dimensional entire solutions verifying the asymptotic conditions u(x, y) → ±1 as x → ±∞ uniformly with respect to y ∊ ℝ.

The multiplicity and concentration of positive solutions are established for the equation −ε p Δ p u + V (z)|u| p-2 u = f(u) in IR N , where V is a positive continuous function and f ∈ C 1 is a function having subcritical and superlinear growth.

We prove the existence of nondecreasing sequences of positive eigenvalues of the homogeneous degenerate quasilinear eigenvalue problem − div(a(x)|▽u| p-2 ▽u) = λb(x)|u| p-2 u, λ > 0 subject to Dirichlet boundary conditions on a bounded domain . The diffusion coefficient a(x) is a function in L 1 loc (Ω) and b(x) is a nontrivial function in L r (Ω) (r depending on a, p and N) and may change sign. We use Ljusternik-Schnirelman theory, minimax theory and the theory of weighted Sobolev spaces to establish our results.

We first obtain an (optimal) regularity condition for entire radial solutions of Monge- Ampère equations and then study the existence of radial solutions of the Semi-geostrophic system which is coupled by a Monge-Ampère and a transport equation.