Positive Radial States and Persistence of Symmetry Breaking for a Spectral Fractional Robin Lane–Emden Equation

Let B \( \subset\ \)R² be the unit disk and let \( A_\beta \) be the positive Robin Laplacian associated with \( a_\beta(u,v)=\int_B \nabla u\cdot\nabla v\,dx+\beta\int_{\partial B}uv\,dS, \qquad \beta>0. \) For \(0<s\le 1\) we study the spectral fractional Lane–Emden problem \( A_\beta^s u=u^p,\qquad u>0\quad\hbox{in }B, \) where the Robin boundary interaction is encoded in the spectral definition of \( A_\beta^s \). The case \( s=1 \) is the classical planar Robin Lane–Emden equation. We first establish variational existence of positive solutions, and of positive radial solutions, in the fractional Sobolev-subcritical range \( p+1<2/(1-s) \). Positivity follows from the positivity-improving Robin heat semigroup and the semigroup formula for \(A_\beta^{-s}\). We then use the classical problem as a nondegenerate endpoint and develop a perturbation theory for \( s\uparrow1 \). In particular, the classical positive radial branch continues to a locally unique fractional radial sheet \( (s,\beta)\mapsto U_{s,\beta} \). The principal result concerns symmetry breaking. For each \( p\ge 8 \), let \(\beta_{p,-}\) and \(\beta_{p,+}\) denote the two distinguished simple first-angular symmetry-breaking parameters of the classical Robin problem established in the published local theory. We prove that, for \(s\) sufficiently close to \( 1 \), these two points persist as two \(C^1\) curves \(\beta_{p,\pm}(s)\) with \(\beta_{p,\pm}(s)\to\beta_{p,\pm}\) as \( s\uparrow1 \). At each point \( (s,\beta_{p,\pm}(s)) \), the fractional linearization has a two-dimensional first-angular kernel in the full space and a one-dimensional kernel in a reflection-invariant subspace. A self-adjoint conjugation of the linearized equation and spectral perturbation theory show that the corresponding simple eigenvalue of the bounded conjugated linearization crosses zero transversally. The Crandall–Rabinowitz theorem then yields local branches of positive nonradial solutions. Hence, for every \( p\ge8 \) and every fractional order sufficiently close to one, uniqueness among all positive solutions fails along parameters converging to each of the two fractional bifurcation curves, while the radial state remains locally unique within the radial class. We also give the extension formulation, the exact disk-scaling law, and explain why a genuinely variable-order model requires a different operator framework.