We consider the electrostatic Born-Infeld energŷ. Moreover, if the norm of ρ is sufficiently small, the minimizer is a weak solution of the associated PDEwith the boundary condition lim |x|→∞ u(x) = 0 and it is of class C 1,α loc (R N ), for some α ∈ (0, 1).2010 Mathematics Subject Classification. 35J93,35Q60, 35B65.

Abstract. We study the asymptotic behavior, as λ → 0, of least energy radial sign-changing solutions u λ , of the Brezis-Nirenberg problemwhere λ > 0, 2 * = 2n n−2 and B 1 is the unit ball of R n , n ≥ 7. We prove that both the positive and negative part u + λ and u − λ concentrate at the same point (which is the center) of the ball with different concentration speeds. Moreover we show that suitable rescalings of u + λ and u − λ converge to the unique positive regular solution of the critical exponent problem in R n .Precise estimates of the blow-up rate of u ± λ ∞ are given, as well as asymptotic relations between u ± λ ∞ and the nodal radius r λ . Finally we prove that, up to constant, λ, where G(x, y) is the Green function of the Laplacian in the unit ball.

In this paper we give necessary and sufficient conditions for the existence of positive radial solutions for a class of fully nonlinear uniformly elliptic equations posed in the complement of a ball in R N , and equipped with homogeneous Dirichlet boundary conditions. 2010 Mathematics Subject Classification. 35J60; 35B33; 34B53.

In this paper, we prove that the Brezis-Nirenberg problem −∆u = |u| p−1 u + ǫuin Ω, u = 0 on ∂Ω,where Ω is a symmetric bounded smooth domain in R N , N ≥ 7 and p = N+2 N−2 , has a solution with the shape of a tower of two bubbles with alternate signs, centered at the center of symmetry of the domain, for all ǫ > 0 sufficiently small. In the case N = 3, a similar result was proved in [9] but only for ǫ ∈ (λ * , λ 1 ) with λ * = λ * (Ω) > 0. Moreover by using a version of the Pohozaev Identity the authors showed that λ * (Ω) = 1 4 λ 1 if Ω is a ball and that no positive solutions exist for ǫ ∈ (0, 1 4 λ 1 ). Note that, by using again Pohozaev Identity, it is easy to check that problem (1.1) has no nontrivial solutions when ǫ ≤ 0 and Ω is star-shaped. Since then, there has been a considerable number of papers on problem (1.1). We briefly recall some of the main ones. Han, in [22], proved that the solution found by Brezis and Nirenberg blows-up at a critical point of the Robin's function as ǫ goes to zero. Conversely, Rey in [30] and in [31] proved that any C 1 − stable critical point of the Robin's function generates a family of positive solutions which blows-up at this point as ǫ goes to zero. After the work of Brezis and Nirenberg, Capozzi, Fortunato and Palmieri [12] showed that for N = 4, ǫ > 0 and ǫ ∈ σ(−∆) (the spectrum of −∆) problem (1.1) has a nontrivial solution. The same holds if N ≥ 5 for all ǫ > 0 (see also [21]).The first multiplicity result was obtained by Cerami, Fortunato and Struwe in [14], in which they proved that the number of nontrivial solutions of (1.1), for N ≥ 3, is bounded below by the number of eigenvalues of (−∆, Ω) belonging to (ǫ, ǫ + S|Ω| −2/N ), where S is the best constant 2010 Mathematics Subject Classification. 35J60 (primary), and 35B33, 35J20 (secondary).

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.