Stability of solutions of a 1-dimensional, p-Laplacian problem and the shape of the bifurcation curve. (July 2020) - Rynne, Bryan P.
Abstract: We consider the p -Laplacian boundary-value problem (1) − φ p ( u ′ ) ′ = λ f ( u ), on ( − 1, 1 ), (2) u ( ± 1 ) = 0, where p > 1 ( p ≠ 2 ), φ p ( z ) : = | z | p − 1 sgn z, z ∈ R, λ ⩾ 0, f : R → R is C 2 and f > 0 on R . Under these conditions the set of solutions ( λ, u ) of (1) –(2) consists of the trivial solution ( λ, u ) = ( 0, 0 ) together with a single (connected) C 2 curve S ⊂ R + × C 0 1 [ − 1, 1 ] ( R + = ( 0, ∞ ) ). Under additional conditions on f the 'shape' of S can be determined. Solutions of (1) –(2) are equilibrium solutions of a related time-dependent, parabolic problem, and in this time-dependent setting the stability of these equilibria is of interest. It will be shown that the stability of solutions on S is determined by the shape of S . This will first be discussed in a general setting, and the results will then be applied to the specific case where S is ' S -shaped'. Finally, similar results will be obtained, for 'generic' λ, without any additional conditions on f .