Morse index of Aubin–Talenti bubbles for 1<p<2

Determine the exact Morse index $i(u_{\lambda,x_0})$ of the AubinTalenti extremals for the critical p-Laplace equation when $1<p<2$ and $p<N$.

Background

For 2<p<N, the paper proves that every AubinTalenti extremal has Morse index one. The proof relies on twice differentiability of the p-energy functional, which fails in the same form for 1<p<2 because the linearized weight becomes singular where the gradient vanishes. Consequently, the exact Morse index in the subquadratic range is not determined.

References

It therefore does not determine $i(u_{\lambda,x_0})$ in that range; to the best of our knowledge, the exact value is not presently available.

— Sharp Liouville thresholds and endpoint rigidity for finite Morse index solutions of the $p$-Laplace Lane--Emden equation  (2610.02751 - Le, 2 Oct 2026) in Final remark of Section 6, following the proof of Theorem 6.3