Band 12, Heft 4

This work deals with bifurcation of positive solutions for some asymptotically linear problems, involving the square root of the Laplacian (−Δ) 1/2 . A simplified model problem is the following: with Ω ⊂ ℝ N a smooth bounded domain, N ≥ 2, λ > 0, m ∈ L ∞ (Ω), m + ≢ 0 and g is a continuous function which is super-linear at 0 and sub-linear at infinity. As a consequence of our bifurcation theory approach we prove some existence and multiplicity results. Finally, we also show an anti-maximum principle in the corresponding functional setting.

We consider the nonlinear Klein-Gordon equation with nonnegative potential, which makes the equation suitable for physical models, and prove the existence of solitary wave solutions with nonvanishing angular momentum and enough largely prescribed charge (c-vortices). This is done by solving the following minimization problem: where x = (y, z) ∈ ℝ 2 × ℝ, k ≠ 0 and H is a suitable subspace of H 1 (ℝ 3 ) × ℝ.

We consider a non-Abelian gauge theory in ℝ 4 equipped with the Minkowski metric, which provides a model for the interaction between a bosonic matter field and a gauge field with gauge group S U(2). We prove the existence of solitary waves which are related to those found for the Klein-Gordon-Maxwell equations.

The system describes pulses in nonlinear fiber couplers. It has the family (U 1+λ , -U 1+λ ), -1 < λ < ∞, of soliton states (that is, homoclinic solutions to the origin), where For λ ≥ 1, the equilibrium (0, 0) is not hyperbolic and therefore the soliton state (U 1+λ , −U 1+λ ) can be qualified as “singular”. In N. Akhmediev and A. Ankiewicz [1], it is observed numerically that a branch of homoclinic solutions bifurcates subcritically at λ = 1 from the family (U 1+λ , −U 1+λ ). The aim of the present paper is to give a rigorous proof of the existence of this bifurcation, as desired in A. Ambrosetti and D. Arcoya [3]. A particular feature of the present problem is that the linearized system at (U 2 , −U 2 ) has a non-constant bounded solution that does not vanish at infinity. Hence the bifurcating homoclinic solutions have a transient “spatial” region where they are well described with the help of this bounded function. Moreover the decay to 0 is governed by two different scales, the larger one originating from the singular aspect of (U 2 , −U 2 ). The existence proof developed here relies on the “broken geodesic” technique to match the inside transient region with the outside region.

We study the existence of a positive solution of a nonlinear elliptic equation where k ≥ 2 and D is a bounded domain domain in R ℓ , ℓ ≥ 1. We give almost necessary and sufficient condition of f(ξ) for the existence of a positive solution, which is inspired by the works of Berestycki-Lions [1] and Berestycki-Gallouët-Kavian [3].

We study the validity of the condition (N) of Lusin for homeomorphisms under minimal assumptions on the integrability of Df. It turns out that the role of grand Sobolev spaces W 1,n) and W b 1,n) is crucial. A discussion of bi-Sobolev maps in the plane and their connections with degenerate elliptic PDEs is provided.

On closed manifolds, gradient and Hessian a priori estimates for fully nonlinear Yamabe problems are known to hold. On manifolds with boundary, gradient estimates are known to hold, while Hessian estimates hold if the prescribed mean curvature is positive. Examples are given here which show that Hessian estimates can fail when the mean curvature is negative.

It is well known that the linear stability of Lagrangian elliptic equilateral triangle homographic solutions in the classical planar three-body problem depends on the mass parameter β = 27(m 1 m 2 + m 2 m 3 + m 3 m 1 )/(m 1 + m 2 + m 3 ) 2 ∈ [0, 9] and the eccentricity ℯ ∈ [0, 1). We are not aware of any existing analytical method which relates the linear stability of these solutions to these two parameters directly in the full rectangle [0, 9] × [0, 1), besides perturbation methods for ℯ > 0 small enough, blow-up techniques for e sufficiently close to 1, and numerical studies. In the recent paper [5], X. Hu, S. Sun and the author introduced a new rigorous analytical method to study the linear stability of these solutions in terms of the two parameters in the full (β, ℯ) range [0, 9] × [0, 1) via the ω-index theory of symplectic paths for ω belonging to the unit circle of the complex plane, and the theory of linear operators. This is an expository article on [5]. In Section 1 of this paper we give a brief review on the history of this stability problem, and describe recent results obtained in [5]. Then in Section 2, after reviewing the ω-index theory in [15], the ω-index decreasing property of the elliptic Lagragian solutions in β for fixed ℯ ∈ [0, 1) proved in [5] is introduced, then the existence of three curves located from left to right in the parameter rectangle [0, 9] × [0, 1) is explained. Among these three curves, two are precisely the −1 degeneracy curves and the third one is the left boundary curve of the hyperbolic subregion in [0, 9] × [0, 1). Then in Section 3, it is explained why the linear stability pattern of such an elliptic Lagrangian solution q β,ℯ changes if and only if the parameter (β, ℯ) passes through one of these three curves. Some open problems on related topics are suggested at the end of this paper.

We will survey three applications of the famous Mountain Pass Lemma by Antonio Ambrosetti and Paul H. Rabinowitz, [6]. Precisely, we will present some results on the following problems: A model of growth and roughening of surfaces, usually called Kardar-Parisi-Zhang model. A supercritical problem involving Hardy-Leray potential with the pole at the boundary. A fourth order nonlinear problem related to epitaxial growth. The common characteristic of all these problems is that we have some extra difficulties in order to apply the Mountain Pass Lemma.

For a semilinear elliptic system of PDE’s which is spatially reversible, we establish the existence of solutions in C 2 (ℝ 2 × T n−2 , ℝ m ) that are heteroclinic in x 2 between a pair of solutions in C 2 (ℝ × T n−1 , ℝ m ) that are each heteroclinic in x 1 . The proof uses minimization arguments.

For an elliptic model equation with supercritical power non-linearity we give a complete description of radial solutions and discuss self-similar blow-up solutions.

The classical Poincaré inequality establishes that for any bounded regular domain Ω ⊂ ℝ N there exists a constant C = C(Ω) > 0 such that In this paper we show that C can be taken independently of Ω when Ω is in a certain class of domains. Our result generalizes previous results in this direction.