Existence of additional stable homogeneous profiles above the Joseph–Lundgren exponent

Determine whether nonzero homogeneous weak solutions of the p-Laplace Lane–Emden equation that are stable outside a ball exist beyond the profiles 7c_s|x|^{-alpha} when q>q_c(N,p).

Background

The paper classifies stable homogeneous profiles at and below the JosephLundgren threshold: none exist for q<q_c(N,p), while at q=q_c(N,p) the only nonzero profiles are 7c_s|x|^{-alpha}. For q>q_c(N,p), the argument no longer excludes other angularly dependent homogeneous profiles because the key coercive quantity has the opposite sign. The existence or nonexistence of such additional profiles therefore remains unresolved.

References

For $q>q_c(N,p)$ the argument stops: the quantity that vanishes at the threshold has the wrong sign, and we do not know whether other homogeneous stable profiles appear.

— 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 Section 4, immediately after the proof of Theorem Homogeneous profiles and the exponent q_c

Consequently, a complete asymptotic classification at the Joseph--Lundgren endpoint would follow once one proves that every blow-down limit is homogeneous. Proposition \ref{prop_qc_omega} shows that this is in fact equivalent to uniqueness of the blow-down: either property forces the full rescaled family to converge to one of $0$ and $\pm c_s|x|{-\alpha}$. This is exactly the point at which the lack of a monotonicity formula for $p\ne2$ remains relevant. We do not claim such a homogeneity theorem here.

— 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 Remark 'What remains at q=q_c' in Section 7.1