Global solution branches for p-Laplacian boundary value problems

We study global continua of positive solutions of the boundary value problem -Δpu=λ(1+uq) in a bounded smooth domain Ω⊂Rn with zero Dirichlet boundary conditions. For subcritical q we show that an unbounded continuum of positive solutions exists with the property that for every λ∈(0,λ*) at least two solutions exist but for λ>λ* no solution exists. In contrast we show for supercritical q that uniqueness holds for small positive λ. We prove our multiplicity result via a topological degree argument and a priori bounds combining recent results of Brock (Proc. Indian Acad. Sci. Math. Sci. 110 (2000) 157-204; Continuous rearrangement and symmetry of solutions of elliptic problems, Habilitation Thesis, Leipzig, 1998, 129 pp.), Damascelli and Pacella (Ann. Scuola Norm. Sup. Pisa Cl. Sci. 26 (1998) 689-707) and Serrin and Zou (Acta Math. 189 (2002) 79-142). The uniqueness result for supercritical q is proven by a Pohožaev-type identity and a new weighted Poincaré inequality of Fleckinger and Takač (Adv. Differential Equations (7) (2002) 951-971).