Ricerc@Sapienza

We consider the following critical elliptic system: {−Δui=μiui3+βui∑j≠iuj2inΩεui=0 on ∂Ωε,ui>0 in Ωεi=1,…,m, in a domain Ωε⊂R4 with a small shrinking hole Bε(ξ0). For μi>0, β<0, and ε>0 small, we prove the existence of a non-synchronized solution which looks like a fountain of positive bubbles, i.e. each component ui exhibits a towering blow-up around ξ0 as ε→0.

Abstract. In this paper we study existence, nonexistence and classification of radial positive
solutions of some weighted fully nonlinear equations involving Pucci extremal operators. Our
results are entirely based on the analysis of the dynamics induced by an autonomous quadratic
system which is obtained after a suitable transformation. This method allows to treat both regular
and singular solutions in a unified way, without using energy arguments. In particular we recover

We study a critical Neumann problem in an unbounded cone Σ_ω:={tx:x∈ω and t>0}, where ω is an open connected subset of the unit sphere S^N−1 in R^N with smooth boundary, N≥3 and 2∗:=2N/N−2. We assume that some local convexity condition at the boundary of the cone is satisfied. If ω is symmetric with respect to the north pole of S^N−1, we establish the existence of a nonradial sign-changing solution.

In this paper, we build infinitely many non-radial sign-changing solutions to the critical problem (P)−Δu=|u|[Formula presented]u, in Ω,u=0, on ∂Ω on the annulus Ω:=x∈RN:a