Liouville type results for a p-Laplace equation with negative exponent

Positive entire solutions of the equation .pu=u.qinRN(N.2) where 1 < p . N, q > 0, are classified via their Morse indices. It is seen that there is a critical power q = q c such that this equation has no positive radial entire solution that has finite Morse index when q > q c but it admits a family of stable positive radial entire solutions when 0 < q . q c. Proof of the stability of positive radial entire solutions of the equation when 1 < p < 2 and 0 < q . q c relies on Caffarelli.Kohn.Nirenberg.s inequality. Similar Liouville type result still holds for general positive entire solutions when 2 < p . N and q > q c. The case of 1 < p < 2 is still open. Our main results imply that the structure of positive entire solutions of the equation is similar to that of the equation with p = 2 obtained previously. Some new ideas are introduced to overcome the technical difficulties arising from the p-Laplace operator.