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• W1969155423 abstract "In this work, we study the existence problem for positive solutions of the Yamabe type equation (Y) Δ u + q ( x ) u − b ( x ) u σ = 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 R m , 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 ( v h ′ ) ′ + A v h = 0 , h ( 0 ) = 1 , h ′ ( 0 ) = 0 , of independent interest." @default.
• W1969155423 created "2016-06-24" @default.
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• W1969155423 date "2015-01-01" @default.
• W1969155423 modified "2023-09-29" @default.
• W1969155423 title "Yamabe type equations with sign-changing nonlinearities on non-compact Riemannian manifolds" @default.
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• W1969155423 doi "https://doi.org/10.1016/j.jfa.2014.10.016" @default.
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