Abstract
We study global continua of positive solutions of the boundary value problem - Δ p u = λ ( 1 + u q ) in a bounded smooth domain Ω ⊂ R n 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).
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