Blow-up solutions for linear perturbations of the Yamabe equation

For a smooth, compact Riemannian manifold (M,g) of dimension N \ge3, we are interested in the critical equation $$\Delta_g u+[N-2/ 4(N-1)S_g +\epsilon h] u=u^{N+2/N-2}$$ in M, u>0 in M, where \Delta_g is the Laplace--Beltrami operator, S_g is the Scalar curvature of (M,g), h\in C^{0,\alpha}(M), and \epsilon is a small parameter.