Compactness results for divergence type nonlinear elliptic equations

Let M be a compact manifold of dimension n >= 2 and 1 < p < n. For a family of functions F-alpha defined on TM, which are p-homogeneous, positive, and convex on each fiber, of Riemannian metrics g(alpha) and of coefficients a(alpha) on M, we discuss the compactness problem of minimal energy type solutions of the equation [GRAPHICS] This question is directly connected to the study of the first best constant A(opt)(alpha) associated with the Riemannian F-alpha-Sobolev inequality [GRAPHICS] Precisely, we need to know the dependence of A(opt)(alpha) under F-alpha and g(alpha). For that, we obtain its value as the supremum on M of best constants associated with certain homogeneous Sobolev inequalities on each tangent space and show that A(opt)(alpha) is attained on M. We then establish the continuous dependence of A(opt)(alpha) in relation to F-alpha and g(alpha). The tools used here are based on convex analysis, blow-up, and variational approach.