Classification and non-degeneracy of positive radial solutions for a weighted fourth-order equation and its application

Abstract

This paper is devoted to radial solutions of the following weighted fourth-order equation div(|x|α∇(div(|x|α∇u)))=|u|2α∗∗−2uinRN,where N≥2, 4−N2<α<2 and 2α∗∗=2NN−4+2α. It is obvious that the solutions of above equation are invariant under the scaling λN−4+2α2u(λx) while they are not invariant under translation when α≠0. We characterize all the solutions to the related linearized problem about radial solutions, and obtain the conclusion of that if α satisfies (2−α)(2N−2+α)≠4k(N−2+k) for all k∈N+ the radial solution is non-degenerate, otherwise there exist new solutions to the linearized problem that “replace” the ones due to the translations invariance. As applications, firstly we investigate the remainder terms of some inequalities related to above equation. Then when N≥5 and 0<α<2, we establish a new type second-order Caffarelli–Kohn–Nirenberg inequality ∫RN|div(|x|α∇u)|2dx≥C∫RN|u|2α∗∗dx22α∗∗,for allu∈C0∞(RN),and in this case we consider a prescribed perturbation problem by using Lyapunov–Schmidt reduction.