Yamabe type equations with a sign-changing nonlinearity, and the prescribed curvature problem (1404.3118v4)

Abstract: In this paper, we investigate the prescribed scalar curvature problem on a non-compact Riemannian manifold $(M, \langle \, , \, \rangle)$, namely the existence of a conformal deformation of the metric $\langle \, , \, \rangle$ realizing a given function $\widetilde s(x)$ as its scalar curvature. In particular, the work focuses on the case when $\widetilde s(x)$ 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 quasi-isometric 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, $\sigma>p-1>0$, $a,b \in L^\infty_{\mathrm{loc}}(M)$ 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 Schr\"odinger type operator $$ Q_V' \ : \ \varphi \longmapsto -\Delta_p \varphi - a(x)|\varphi|^{p-2}\varphi. $$ In particular, we prove sharp Hardy-type inequalities in some geometrically relevant cases, notably for minimal submanifolds of the hyperbolic space.