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On the Behaviour of the First Eigenfunction of the p ‐Laplacian Near its Critical Points
Author(s) -
GarcíaMelián Jorge
Publication year - 2003
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609303001966
Subject(s) - convexity , eigenfunction , mathematics , mathematics subject classification , critical point (mathematics) , laplace operator , symmetry (geometry) , combinatorics , pure mathematics , mathematical analysis , eigenvalues and eigenvectors , geometry , physics , quantum mechanics , financial economics , economics
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).