Positive entire solutions of the equation Δ_{p}u=u^{-q} in R^{N} (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.

Zong Ming GUO,Lin Feng MEI. Liouville Type Results for a p-Laplace Equation with Negative Exponent[J]. Acta Mathematica Sinica, English Series, 2016, 32(12): 1515-1540.

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