Bifurcation along curves for the p-Laplacian with radial symmetry

Abstract

We study the global structure of the set of radial solutions of a nonlinear Dirichlet problem involving the p-Laplacian with p>2, in the unit ball of $R^N$, $N \ges 1$. We show that all non-trivial radial solutions lie on smooth curves of respectively positive and negative solutions and bifurcating from the line of trivial solutions. This involves a local bifurcation result of Crandall-Rabinowitz type, and global continuation arguments relying on monotonicity properties of the equation. An important part of the analysis is dedicated to the delicate issue of differentiability of the inverse p-Laplacian. We thus obtain a complete description of the global continua of positive/negative solutions bifurcating from the first eigenvalue of a weighted, radial, p-Laplacian problem, by using purely analytical arguments, whereas previous related results were proved by topological arguments or a mixture of analytical and topological arguments. Our approach requires stronger hypotheses but yields much stronger results, bifurcation occuring along smooth curves of solutions, and not only connected sets.