On Pleijel-type nodal domain bounds for the $p$-Laplacian

We provide an upper estimate \`a la Pleijel on the asymptotic number of nodal domains for eigenfunctions corresponding to the cogenus eigenvalues $\{\lambda_k(p;\Omega)\}$ of the $p$-Laplacian in a bounded domain $\Omega$, and identify regimes when the number of nodal domains of the $k$-th eigenfunction is less than $k$ as $k \to +\infty$. As auxiliary results, which also have independent interest, we provide a useful characterization of the cogenus eigenvalues implying their continuity with respect to $p$, justify the Weyl law, and prove the inequality $\lambda_2(p;B) \leq \dots \leq \lambda_{N+1}(p;B) \leq \lambda_\ominus(p)$ in an $N$-dimensional ball $B$, where $\lambda_\ominus(p)$ is an eigenvalue whose eigenfunction has a central section of $B$ as its nodal set.