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

Abstract

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