Stability of solutions of a 1-dimensional, [formula omitted]-Laplacian problem and the shape of the bifurcation curve

Abstract

We consider the p-Laplacian boundary-value problem (1)−φp(u′)′=λf(u),on (−1,1),(2)u(±1)=0, where p>1 (p≠2), φp(z):=|z|p−1sgnz, z∈R, λ⩾0, f:R→R is C2 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) C2 curve S⊂R+×C01[−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.