Non-Linear Elliptic Equations on Riemannian Manifolds with the Sobolev Critical Exponent

Abstract

Given (M n ,g) a Riemannian compact manifold, without boundary of dimension n and a function q ∈ L∞ (M, ℝ), we study by variational methods the following class ofequations: $$\left\{{\begin{array}{*{20}{c}}{(-\Delta+q)u={u^{\frac{{n+2}}{{n-2}}}}}\\{u>0on{M^n}}\end{array}}\right.$$ Extending a method devised in [7] and [10], which enables to find solutions to such equations although the corresponding variational problem fails the Palais-Smale condition, we show that this method works also in this new case, where the back-ground metric is not flat. Our result provides, in particular, a new proof of the existence of a solution to the Yamabe conjecture in dimensions 3, 4, and 5. For the locally conformally flat case (n ≥ 6), this method also works as has been established elsewhere ([11]).