Yamabe type equations with a sign-changing nonlinearity, and the prescribed curvature problem

作者：Bianchini Bruno; Mari Luciano^*; Rigoli Marco

来源：Journal of Differential Equations, 2016, 260(10): 7416-7497.

DOI：10.1016/j.jde.2016.01.031

• Yamabe equation
• Schrodinger operator
• Subcriticality
• p-Laplacian
• Spectrum
• Prescribed curvature

In this paper, we investigate the prescribed scalar curvature problem on a non-compact Riemannian manifold (M, <,>), namely the existence of a conformal deformation of the metric <,> realizing a given s) over tilde (x) as its scalar curvature. In particular, the work focuses on the case when (s) over tilde (x) changes sign. Our main achievement are two new existence results requiring minimal assumptions on the underlying manifold, and ensuring a control on the stretching factor of the conformal deformation in such a way that the conformally deformed metric be bi-Lipschitz equivalent to the original one. The topological-geometrical requirements we need are all encoded in the spectral properties of the standard and conformal Laplacians of M. Our techniques can be extended to investigate the existence of entire positive solutions of quasilinear equations of the type
Delta(p)u + a(x)u(p-1)-b(x)u(sigma) = 0
where Delta(p) is the p-Laplacian, sigma > p - 1 > 0, a, b is an element of L-loc(infinity)(M) and b changes sign, and in the process of collecting the material for the proof of our theorems, we have the opportunity to give some new insight on the subcriticality theory for the Schrodinger type operator
Q'(V) : phi -> -Delta(p)phi-a(x)vertical bar phi vertical bar(p-2)phi.
In particular, we prove sharp Hardy-type inequalities in some geometrically relevant cases, notably for minimal submanifolds of the hyperbolic space.

• 出版日期2016-5-15