Volume 11, Issue 3

Uniqueness of positive radial solutions decaying at infinity is proved for a class of semilinear elliptic equations on ℝ 2 . Complementary results for the same kind of equations were obtained in the early 90’s, on ℝ N with N ≥ 3, and in finite balls of ℝ N with N ≥ 2. The new result presented here plays a crucial role in the global bifurcation problem, previously studied by the author.

The results presented here were motivated by several recent papers on singular boundary value problems for semilinear elliptic equations with convection terms. We present extensions which cover singular nonlinear equations (mainly equations involving the p−Laplacian) containing convection terms. The results obtained are proved using sub- and supersolution theorems (motivated by the results in [18, 19, 20, 23]) and the construction of a well-ordered pair of such using a principal eigenfunction of the p−Laplacian.

We investigate the local and global well-posedness for a class of nonlinear Schrödinger systems with an external time independent electromagnetic field and nonlocal nonlinearities. New conditions related to the growth of the nonlocal term are detected which allow the solvability of the problem.

We give sufficient conditions for the existence of almost periodic solutions of the secondorder differential equation u′′(t) = f (u(t)) + e(t) on a Hilbert space H, where the vector field f : H → H is monotone, continuous and the forcing term e : ℝ → H is almost periodic. Notably, we state a result of existence and uniqueness of the Besicovitch almost periodic solution, then we approximate this solution by a sequence of Bohr almost periodic solutions.

We introduce a hierarchy of evolution equations based on the Sturm-Liouville equation −(pφ′)′ + qφ = λyφ. Our hierarchy includes the Korteweg-de Vries (K-dV) and the Camassa-Holm (CH) hierarchy. We determine a class of solutions of the hierarchy which are of algebro-geometric type. The initial condition of such a solution is drawn from a finite-gap isospectral class of the Sturm-Liouville equation.

This paper is a preliminary work on Heisenberg group domains, devoted to the study of the Green’s function for the Kohn Laplacian on domains far away from the set of characteristic points. We give some estimates of the Green’s function, its regular part and their derivatives analogous to those proved by A. Bahri, Y.Y. Li, O. Rey in [1], and O. Rey in [16] for Euclidean domains. While the study of such functions on the set of characteristic points of the given domain will be discussed in a forthcoming paper.

By means of a continuation argument, we prove the existence of at least one increasing heteroclinic solution to a scalar equation of the kind ẍ = a(t)V′(x), where V is a non-negative double well potential, and a(t) is a positive, measurable coefficient, which is definitively monotone with respect to |t|, converges to l as |t| diverges and fulfils one of the two following assumptions: a(t) ≥ l everywhere, or a(t) − l converges to 0, as |t| → +∞, more slowly than a suitable exponential term.

The b−family of equations governs evolution of shallow water waves with the horizontal velocities in different depths related to the parameter b. Using the method of characteristics, we obtain the results of the exponential decay properties of the strong solutions for the b−family of equations. In particular, we show that a non-trivial strong solution with compact initial value can not be compactly supported at any later time.

We consider periodic second-order equations having an ordered pair of lower and upper solutions and show the existence of asymptotic trajectories heading towards the maximal and minimal periodic solutions which lie between them.

We examine the elliptic system given by where λ, γ are positive parameters and where Ω is a smooth bounded domain in ℝ N . Let U denote the parameter region (λ, γ) of strictly positive parameters where (P) λ,γ has a smooth solution and let Υ denote the boundary of U. We show that the extremal solution (u∗, v∗) associated with (λ∗, γ∗) ∈ Υ is smooth provided that 3 ≤ N ≤ 9 and

We investigate the continuity of solutions for general nonlinear parabolic equations with non-standard growth near a nonsmooth boundary of a cylindrical domain. We prove the Wiener Criterion for the regularity of a boundary point.