Bifurcation and Concentration of Radial Solutions of a Nonlinear Degenerate Elliptic Eigenvalue Problem

Abstract

Abstract We consider the nonlinear boundary-value problem in ℝ N (N ≥ 3), where C ∊ C 1 ([0, 1]) with C(r) > 0 for all r ∊ [0, 1]; C(0) = 0 and and, for some T > 0; f ∊ C 1 ([-T, T]) is an odd function that is strictly concave on [0, T] with f(0) = f(T) = 0 and f′(0) = 1. We prove that there is vertical bifurcation of radial solutions at every in the sense that there exists a sequence {(λ, w n )} of solutions such that These solutions concentrate at 0 in the sense that w n (0) = T for all n but w n converges uniformly to zero on all compact subsets that do not contain zero.