In this paper we continue the analysis of the blow-up of low energy sign-changing solutions of semilinear elliptic equations with critical Sobolev exponent, started in [M. Ben Ayed, K. El Mehdi, F. Pacella, Blow-up and nonexistence of sign-changing solutions to the Brezis-Nirenberg problem in dimension three, Ann. Inst. H. Poincaré Anal. Non Linéaire, in press]. In addition we prove axial symmetry results for the same kind of solutions in a ball.

In this paper we study low energy sign changing solutions of the critical exponent problem (P λ): − u = u 5 + λu in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in R 3 and λ is a real positive parameter. We make a precise blow-up analysis of this kind of solutions and prove some comparison results among some limit values of the parameter λ which are related to the existence of positive or of sign changing solutions.

In this paper, we consider the problem of prescribing the scalar curvature under minimal boundary conditions on the standard four dimensional half sphere. We provide an Euler-Hopf type criterion for a given function to be a scalar curvature for some metric conformal to the standard one. Our proof involves the study of critical points at infinity of the associated variational problem.

In this paper we make the analysis of the blow up of low energy sign-changing solutions of a semilinear elliptic problem involving nearly critical exponent. Our results allow to classify these solutions according to the concentration speeds of the positive and negative part and, in high dimensions, lead to complete classification of them. Additional qualitative results, such as symmetry or location of the concentration points are obtained when the domain is a ball.