An identity for a quasilinear ODE and its applications to the uniqueness of solutions of BVPs

The following boundary value problem(1.1) ( φ p ( u ′ ) ) ′ + a ( x ) f ( u ) = 0 , x 0 < x < x 1 , (1.2) u ( x 0 ) = u ( x 1 ) = 0 , is considered, where φ p ( s ) = | s | p − 2 s , p > 1 , a ∈ C 1 [ x 0 , x 1 ] , a ( x ) > 0 for x ∈ [ x 0 , x 1 ] , and f ∈ C 1 ( R ) . An identity for solutions of (1.1) and its linearized equation is derived. Some applications of the identity to uniqueness of solutions of problem (1.1)–(1.2) are presented. Non-uniqueness examples for problem (1.1)–(1.2) are also established. Moreover the results obtained here are applied to the study of radially symmetric solutions of the Dirichlet problem for elliptic equations in annular domains.