In this paper we study the asymptotic and qualitative properties of least energy radial signchanging solutions of the fractional Brezis-Nirenberg problem ruled by the s-Laplacian, in a ball of R n , when s ∈ (0, 1) and n > 6s. As usual, λ is the (positive) parameter in the linear part in u. We prove that for λ sufficiently small such solutions cannot vanish at the origin, we show that they change sign at most twice and their zeros coincide with the sign-changes. Moreover, when s is close to 1, such solutions change sign exactly once. Finally we prove that least energy nodal solutions which change sign exactly once have the limit profile of a "tower of bubbles", as λ → 0 + , i.e. the positive and negative parts concentrate at the same point with different concentration speeds.

Abstract.We investigate the problem of entire solutions for a class of fourth-order, dilation invariant, semilinear elliptic equations with powertype weights and with subcritical or critical growth in the nonlinear term. These equations define noncompact variational problems and are characterized by the presence of a term containing lower order derivatives, whose strength is ruled by a parameter λ. We can prove existence of entire solutions found as extremal functions for some Rellich-Sobolev type inequalities. Moreover, when the nonlinearity is suitably close to the critical one and the parameter λ is large, symmetry breaking phenomena occur and in some cases the asymptotic behavior of radial and nonradial ground states can be somehow described.Mathematics Subject Classification. 26D10, 47F05.

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We study the motion of a charged particle under the action of a magnetic field with cylindrical symmetry. In particular we consider magnetic fields with constant direction and with magnitude depending on the distance r from the symmetry axis of the form 1 + Ar −γ as r → ∞, with A = 0 and γ > 1. With perturbative-variational techniques, we can prove the existence of infinitely many trajectories whose projection on a plane orthogonal to the direction of the field describe bounded curves given by the superposition of two motions: a rotation with constant angular speed at a unit distance about a point which moves along a circumference of large radius ρ with a slow angular speed ε. The values ρ and ε are suitably related to each other. This problem has some interest also in the context of planar curves with prescribed curvature.