In this paper, we consider the Lane–Emden problem [Formula: see text] where Ω is a bounded domain in ℝN and p > 2. First, we prove that, for p close to 2, the solution is unique once we fix the projection on the second eigenspace. From this uniqueness property, we deduce partial symmetries of least energy nodal solutions. We also analyze the asymptotic behavior of least energy nodal solutions as p goes to 2. Namely, any accumulation point of sequences of (renormalized) least energy nodal solutions is a second eigenfunction that minimizes a reduced functional on a reduced Nehari manifold. From this asymptotic behavior, we also deduce an example of symmetry breaking. We use numerics to illustrate our results.

If Ω ⊂ R n is a smooth bounded domain and q ∈ (0, n n−1 ) we consider the Poincaré-Sobolev inequalityfor every u ∈ BV(Ω) such that Ω |u| q−1 u = 0. We show that the sharp constant is achieved. We also consider the same inequality on an ndimensional compact Riemannian manifold M . When n ≥ 3 and the scalar curvature is positive at some point, then the sharp constant is achieved. In the case n ≥ 2, we need the maximal scalar curvature to satisfy some strict inequality.2000 Mathematics Subject Classification. Primary: 46E35; Secondary: 26B30, 26D15, 35J62.

In this article, we give a simple proof of the result due to Lin and Wang ensuring the foliated Schwarz symmetry of the extremal functions for the Caffarelli-Kohn-Nirenberg inequalities. This new proof uses a direct and powerful method due to Bartsch, Weth and Willem using polarizations.

Using continuation theorems of Leray-Schauder degree theory, we obtain existence results for the first order quasilinear boundary value problem À fðuÞ Á 0 ¼ f ðt; uÞ; uðTÞ ¼ buð0Þ;where f : R ! ðÀa; aÞ is an homeomorphism such that fð0Þ ¼ 0 and f : ½0; T  R ! R is a continuous function, a and T being positive real numbers and b some non zero real number.

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