Volume 6, Issue 2

The paper contains some results concerning the existence of a-symmetric solutions to a class of autonomous nonlinear Schrödinger systems, coupled through a small parameter.

We look for standing waves of nonlinear Schrödinger equation coupled with Maxwell’s equations. We use the variational formulation introduced by Benci and Fortunato in 1992 for studying an eigenvalue problem for the Schrödinger-Maxwell system in bounded domains. We establish the existence of multiple standing waves both in the homogeneous and the non-homogeneous cases by means of the fibering method introduced by Pohozaev.

In this paper, we are concerned with the existence of solutions of the problem: where Ω ⊂ ℝ N is an exterior domain and f″(0) = 0.

We establish a priori bounds for positive solutions of semilinear elliptic systems of the form where Ω is a bounded and smooth domain in ℝ 2 . We obtain results concerning such bounds when f and g depend exponentially with respect to u and v.

We consider the nonlinear boundary-value problem in ℝ N (N ≥ 3), where C ∊ C 1 ([0, 1]) with C(r) > 0 for all r ∊ [0, 1]; C(0) = 0 and and, for some T > 0; f ∊ C 1 ([-T, T]) is an odd function that is strictly concave on [0, T] with f(0) = f(T) = 0 and f′(0) = 1. We prove that there is vertical bifurcation of radial solutions at every in the sense that there exists a sequence {(λ, w n )} of solutions such that These solutions concentrate at 0 in the sense that w n (0) = T for all n but w n converges uniformly to zero on all compact subsets that do not contain zero.

The problem with boundary Dirichlet zero data is considered in an exterior domain Ω ⊂ ℝ N . Assuming that, as ε →0, a ε 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 a ε (x) has a suitable behaviour at infinity, the existence of another positive solution is shown.

We review the classical Principle of the Least Action in a general context where the Hamilton functionH is possibly non-convex. We show how the van Groesen [6] principle follows as a particular case where H is hyperregular and of homogeneous type. Homogeneous scalar field spacetimes in spherical symmetry are derived as an application.

The aim of this paper is to prove some existence and multiplicity results for functionals of type J(u) = ∫ Ω ­ A(x, u)|▽u| 2 dx - ∫ Ω ­ G(x, u)dx, u 2 D ∊ H 0 1 (Ω), with bounded domain Ω in ℝ N . Since, in general, J is not Gâteaux differentiable in D, we study its restriction on the Banach space X = H 0 1 (Ω) ∩ L ∞ (Ω) and apply some abstract existence and multiplicity theorems involving a variant of condition (C) below.

In this paper we study the geodesic connectedeness of a class of Lorentzian manifolds with boundary, extending the result of [13]. The results are obtained using variational methods.

We study superlinear elliptic boundary value problems with perturbed symmetries in domains which are invariant under the action of a group G. We give conditions on the growth of the nonlinearity which guarantee the existence of infinitely many G-invariant solutions. These conditions improve those obtained by Bahri and Lions (1988) and Bolle, Ghoussoub and Tehrani (2000) if the domain contains a G-orbit of large enough dimension.

In this paper we discuss the existence of normalized solutions of nonlinear elliptic PDEs in gaps of the essential spectrum of the corresponding differential operator. This issue can he of interest, for instance, in nonlinear optics or in crystalline models with impurities.