Yamabe type equations with sign-changing nonlinearities on non-compact Riemannian manifolds

Abstract

In this work, we study the existence problem for positive solutions of the Yamabe type equation(Y)Δu+q(x)u−b(x)uσ=0,σ>1, on complete manifolds possessing a pole, the main novelty being that b(x) is allowed to change signs. This relevant class of PDEs arises in a number of different geometric situations, notably the (generalized) Yamabe problem, but the sign-changing case has remained basically unsolved in the literature, with the exception of few special cases. This paper aims at giving a unified treatment for (Y), together with new, general existence theorems expressed in terms of the growth of |b(x)| at infinity with respect to the geometry of the manifold and to q(x). We prove that our results are sharp and that, even for Rm, they improve on previous works in the literature. Furthermore, we also detect the asymptotic profile of u(x) as x diverges, and a detailed description of the influence of q(x) and of the geometry of M on this profile is given. The possibility to express the assumptions in an effective and simple way also depends on some new asymptotic estimates for solutions of the linear Cauchy problem (vh′)′+Avh=0, h(0)=1, h′(0)=0, of independent interest.