Some Remarks on the Fučı́k Spectrum of the p-Laplacian and Critical Groups

By Norman E. Dancer, K. W. U. O. Sydney, Nsw, Australia., Florida Institute of Technology, Melbourne., Fl, A. U.S. 2000

59 citations recorded by the source corpus.

Let be a bounded domain in n n ≥ 1. For 1 < p < ∞, the Fucik spectrum of the p-Laplacian on W 1 p 0 is defined as the set p of those points a b ∈ 2 such that − pu = a u+ p−1 − b u− p−1 u ∈ W 1 p 0 (1.1) has a nontrivial solution. Here pu = div ( ∇u p−2∇u) and u± = max ±u 0 . It is known that the first eigenvalue λ1 of − p on W 1 p 0 is positive, simple, and admits a positive eigenfunction φ1 ∈ W 1 p 0 ∩C1 (see Lindqvist [8]), so p contains the two lines λ1 × and × λ1. A first nontrivial curve C2 in p through λ2 λ2 , where λ2 is the second eigenvalue of − p, was recently constructed and variationally characterized by a

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6e9d614ac4c605655b8ad365c59d5147524ba277

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Multiple solutions for coercive p-Laplacian equations (2006, background): “In [6], this topological property of Ob and the limits (1.”
Nontrivial critical groups in $p$-Laplacian problems via the Yang index (2003, unknown): “The only results that the author is aware of in this case are those of Dancer and the author himself [ 8 ] showing that”
Sign-changing and multiple solutions for the $p$-Laplacian (2002, background): “As for regularity results up to the boundary (C1,α-regularity) we refer, for example, to Giaquinta and Giusti [16], Giaquinta [15], Liu and Barrett [21, 22 ], Lieberman [18, 19], or Manfredi [24].”
p-LAPLACIAN PROBLEMS WHERE THE NONLINEARITY CROSSES AN EIGENVALUE (2005, background): “[6] N. Dancer and K. Perera, Some remarks on the Fuč́ık spectrum of the p-Laplacian and critical groups, J. Math.”
Morse Theoretic Aspects of $p$-Laplacian Type Operators (2010, background): “However, only a few papers have used Morse theoretic methods to study such problems; see, e.g., Vannella [130], Cingolani and Vannella [29, 31], Dancer and Perera [40], Liu and Su [74], Jiu and Su [58], Perera [98, 99, 100], Bartsch and Liu [15], Jiang [57], Liu and Li [75], Ayoujil and El Amrouss [”
On sign-changing and multiple solutions of the p-Laplacian (2003, background): “[7,9,12,15,16,18,20,27]), but by our knowledge there is no results for sign-changing solutions about p-Laplacian until now.”
Anisotropic nonlinear Neumann problems (2011, unknown): “We mention the works of Bartsch–Liu [6], Carl–Perera [8], Dancer–Perera [12], Filippakis–Kristaly– Papageorgiou [20], Gasiński–Papageorgiou [23], Liu–Liu [30], Papageorgiou–Papageorgiou [34,35] and Zhang–Chen–Li [38].”