The zero scalar curvature Yamabe problem on noncompact manifolds with boundary

Let $(M^n,g),n\ge 3$ be a noncompact complete Riemannian manifold with compact boundary and $f$ a smooth function on $\mathrm{\partial }M$. In this paper we show that for a large class of such manifolds, there exists a metric within the conformal class of $g$ that is complete, has zero scalar curvature on $M$ and has mean curvature $f$ on the boundary. The problem is equivalent to finding a positive solution to an elliptic equation with a non-linear boundary condition with critical Sobolev exponent.