The Sharp First-Angular Symmetry-Breaking Threshold for the Planar Lane–Emden Equation with Robin Boundary Conditions

This note sharpens the first-angular symmetry-breaking analysis for the planar Lane--Emden equation with positive Robin boundary conditions. We prove the existence of a unique sharp transition exponent\[p_{\mathrm{sb}}\in(5,8).\]For \(1 p_{\mathrm{sb}}\), it has exactly two simple zeros. As a consequence, for every \(p>p_{\mathrm{sb}}\), the spectral simplicity and transversality properties established in the author's previous work allow the Crandall--Rabinowitz theorem to produce two local branches of positive nonradial solutions bifurcating from the unique positive radial branch. The threshold is characterized by an exact scalar shooting equation obtained from the phase-plane reduction. Numerical shooting, included only for orientation and not used in the proof, gives\[p_{\mathrm{sb}}\approx 6.3912867717,\qquad\beta_{*}\approx 0.7539990637.\] This result determines the sharp threshold for the first-angular symmetry-breaking mechanism in the planar pure-power Robin Lane--Emden problem. It does not assert global uniqueness of all positive solutions below \(p_{\mathrm{sb}}\). This work is a follow-up to:R.~Zeraoulia,``Symmetry Breaking for the Planar Lane--Emden Equation with Robin Boundary Conditions,''\emph{Nonlinear Analysis: Real World Applications},Vol.~95, Article~104752,\[\texttt{https://doi.org/10.1016/j.nonrwa.2026.104752}.\]