A Stability Result for Extremal Functions for Some Sharp Sobolev Inequalities with Critical Exponent

Abstract

This Paper is devoted to a proof of a stability result on the solutions of the equation on an open bounded set of , where p∗stands for the critical exponent for the Sobolev embedding of W ^1-pinto L^k spaces, and Δ ^p for the p-Laplacian operator. In particular, we use the concentration-compactness theorem of P.L.Lions in order to prove that, under some assumptions on the data, the solutions cannot converge to 0.

Keywords:

• Sobolev inequalities
• best constant problems
• variational methods
• nonlinear elliptic PDE
• p-Laplacian operator

AMS:

• 35J20
• 35J60
• 35J70
• 49K20