Symmetry Of Extremal Functions In Moser-trudinger Inequalities And A Henon Type Problem In Dimension Two

Symmetry of extremal functions in Moser–Trudinger inequalities and a H´ enon type problem in dimension two.∗ Denis Bonheure1,† , Enrico Serra2 , Massimo Tarallo2 1 Institut de Math´ematique pure et appliqu´ee, Universit´e catholique de Louvain, Chemin du Cyclotron 2, 1348 Louvain–la–Neuve, Belgium 2 Dipartimento di Matematica, Universit`a di Milano, Via Saldini 50, 20133 Milano, Italy July 12, 2007 Mathematics subject classification: 35J65, 46E35. Keywords: Moser–Trudinger inequality, exponential nonlinearities, symmetry breaking, H´enon type equation Abstract In this paper, we analyze the symmetry properties of maximizers of a H´enon type functional in dimension two. Namely, we study the symmetry of the functions that realize the maximum Z ³ ´ 2 sup eγu − 1 |x|α dx, u∈H 1 (Ω) ||u||≤1 Ω where Ω is the unit ball of R2 and α, γ > 0. We identify and study the limit functional Z ³ ´ 2 eγu − 1 dσ, sup u∈H 1 (Ω) ||u||≤1 ∂Ω which is the main ingredient to describe the behavior of maximizers as α → ∞. We also consider the limit functional as α → 0 and the properties of its maximizers. ∗ This research was supported by MIUR Project “Variational Methods and Nonlinear Differential Equations”. † Research supported by the F.S.R. - F.N.R.S. 1 1 Introduction In this paper we address the question of symmetry properties of extremal functions for certain maximization problems related to Moser–Trudinger type inequalities. These are the natural extension to dimension two of classical inequalities involving the Sobolev space H 1 that hold in dimension n ≥ 3. Although the validity of these inequalities has long been established, the question of symmetry properties of extremal functions is, for some problems, a rather recent research topic which has generated considerable efforts in the last few years (see