On the Behaviour of the First Eigenfunction of the p ‐Laplacian Near its Critical Points

In this paper, the behaviour of the positive eigenfunction φ of – Δ p u =   λ| u |p − 2 u in Ω u |∂Ω = 0, p > 1, is studied near its critical points. Under some convexity and symmetry assumptions on Ω, φ is seen to have a unique critical point at x = 0; also, the behaviour of both φ and ∇φ is determined nearby. Positive solutions u to some general problems −Δ p u = f ( u ) in Ω, u |∂Ω = 0, are also considered, with some convexity restrictions on u . 2000 Mathematics Subject Classification 35B05 (primary), 35J65, 35J70 (secondary).