Boundary Vanishing Orders and Asymptotic Expansions for Planar Harmonic Functions with a Sublinear Robin Condition

We study boundary vanishing orders and asymptotic expansions of planar harmonic functions satisfying the sublinear Robin condition ${\mathrm{\partial }}_{\nu }u=\lambda u-\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u,$ with $\lambda \ge 0$ and $0<p<1$. For nontrivial global solutions on bounded connected Lipschitz domains, we bound the number of nodal domains by the number of Steklov eigenvalues strictly below $\lambda$ and establish finite vanishing order at every boundary point. The boundary scaling gives the critical exponent $\frac{1}{1-p}$. On $C^1$ boundaries, every positive subcritical vanishing order is an integer, and the leading term is a unique homogeneous Neumann profile in conformal coordinates. At a noninteger critical order, the rescaled solution converges in normalized graph coordinates to an explicit homogeneous profile determined by $p$ up to sign. Every supercritical bulk vanishing order $m$ is an integer, with a unique homogeneous Dirichlet profile in conformal coordinates. The boundary trace has vanishing order $\frac{m-1}{p}>m$. Integer critical vanishing orders are excluded if the boundary is $C^1$ with Dini mean oscillation.