For every admissible integer n≥ 12 and every exponent p satisfying p_JL(n)<p<p_JL(n-1), every positive smooth entire solution u:R^n→(0,∞) of -Delta u=u^p that is stable, meaning that int_R^n(|nabla varphi|^2-pu^p-1varphi^2),dx≥ 0 for every varphi∈ C_c^∞(R^n), is radially symmetric.

Chan–Wei radiality conjecture

Chan and Wei formulated the conjecture in 2018: positive stable entire solutions of the Lane–Emden equation should be radially symmetric when $p_{\mathrm{JL}}(n)<p<p_{\mathrm{JL}}(n-1)$ . The new construction challenges this conclusion for specified dimensions and exponents.

Known results

Chan and Wei, 2018: formulated the radiality conjecture.
Liu–Wang–Wei–Wu: proved partial results under additional reflection symmetry and a stronger restriction on $p$ .

August 2026 counterexample claim

Shibing Chen, Yong Liu, Juncheng Wei, and Wen Yang state a theorem constructing positive, smooth, nonradial stable entire solutions for admissible $n\ge 12$ and $p$ sufficiently close below $p_{\mathrm{JL}}(n-1)$ , with $p_{\mathrm{JL}}(n)<p<p_{\mathrm{JL}}(n-1)$ . Listed dimensions below $150$ include $26,27,49,50,51,80,81,82,83,120,121,122,123$ . The preprint claims this disproves the conjecture in every admissible dimension, but no independent verification is reported.

Current status (as of August 2026): The conjecture is claimed false for the stated admissible dimensions and parameter range, while the preprint’s construction remains independently unverified and broader formulations remain open.

Chan–Wei radiality conjecture

Progress summary

Primary source

Additional references

Solutions 0