On a class of singularly perturbed elliptic equations in divergence form: existence and multiplicity results

In this paper, we perform a fine blow-up analysis for a boundary value elliptic equation involving the critical trace Sobolev exponent related to the conformal deformation of the metrics on the standard ball, namely the problem of prescribing the boundary mean curvature. From this analysis some a priori estimates in low dimension are obtained. With these estimates, we prove the existence of at least one solution when an index-counting formula associated to the prescribed mean curvature is different from zero.