Simple bifurcation and global curves of solutions of p-Laplacian problems with radial symmetry

Abstract

We consider the boundary-value problem(1)−(rN−1ϕp(u′(r)))′=λrN−1f(r,u(r)),r∈(0,1),(2)BCN(u)=(0,0), where N⩾1 is an integer, p∈R satisfies 2≠p>1, ϕp(s):=|s|p−1signs, s∈R, λ⩾0, andBCN(u)={(u(0),u(1)),if N=1,(u′(0),u(1)),if N>1. The case N=1 is a standard, 1-dimensional, Dirichlet Sturm–Liouville problem, whereas the case N>1 arises when searching for radially symmetric solutions of a PDE Dirichlet p-Laplacian problem on the unit ball in RN.Under suitable additional assumptions on f we obtain a ‘simple bifurcation’ theorem giving C1 curves of solutions bifurcating from trivial solutions at (weighted) eigenvalues of the p-Laplacian, and also obtain global C1 curves of positive solutions. Similar results have been obtained before, but with the restriction that p>2 for various differentiability reasons related to the p-Laplacian. We extend these results to all 2≠p>1.The crux of the proofs lies in extending various differentiability results for solution operators for the p-Laplacian to 2≠p>1 which had previously been obtained for p>2.