Weighted Neumann-to-Steklov limits for nonlinear eigenvalues and trace constants

We study a nonlinear Neumann-to-Steklov limit generated by a family of interior weights concentrating at the boundary. On a class of admissible possibly irregular domains obtained from the unit ball by trace-compatible Sobolev homeomorphisms, we consider the first nontrivial weighted $(p,q)$-Neumann eigenvalue with respect to a concentrating bulk weight $\gamma_a$. We prove that, as $a\to0$, these eigenvalues converge to the corresponding weighted $(p,q)$-Steklov eigenvalue with boundary weight $\beta$. Moreover, normalized minimizers converge, up to subsequences, strongly in $W^{1,p}$ to Steklov minimizers. Equivalently, the best constants in the weighted Poincar\'e inequalities converge to the best constants in the weighted trace inequalities; in fact, a quantitative convergence estimate is obtained in the subcritical trace range.