Electron. J. Diff. Equ., Vol. 2012 (2012), No. 124, pp. 1-17.

Bifurcation along curves for the p-Laplacian with radial symmetry

Francois Genoud

We study the global structure of the set of radial solutions of a nonlinear Dirichlet eigenvalue problem involving the p-Laplacian with p>2, in the unit ball of $\mathbb{R}^N$, $N \geq 1$. We show that all non-trivial radial solutions lie on smooth curves of respectively positive and negative solutions, bifurcating from the first eigenvalue of a weighted p-linear problem. Our approach 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, and holds for all p>1.

Submitted March 7, 2012. Published July 31, 2012.
Math Subject Classifications: 35J66, 35J92, 35B32.
Key Words: Dirichlet problem; radial p-Laplacian; bifurcation; solution curve.

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François Genoud
Department of Mathematics and
the Maxwell Institute for Mathematical Sciences
Heriot-Watt University
Edinburgh EH14 4AS, United Kingdom
email: F.Genoud@hw.ac.uk

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