Bifurcation along curves for the p-Laplacian with radial symmetry

TL;DR

This paper analyzes the global structure of radial solutions to a nonlinear p-Laplacian Dirichlet problem, revealing smooth bifurcation curves from trivial solutions and providing a detailed analytical description of solution continua.

Contribution

It offers a new analytical approach to describe bifurcation curves of solutions, improving upon previous topological methods by ensuring smoothness and stronger results.

Findings

Radial solutions form smooth bifurcation curves from trivial solutions.

Global continua of solutions are fully characterized analytically.

Bifurcation occurs along smooth solution curves, not just connected sets.

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.