arXiv:2611.03971v1 Announce Type: new Abstract: Let $\Omega\subset\mathbb{R}^N$ be a bounded open connected set, and let $q\in(1,2)$. We consider the sublinear $q$-eigenvalue problem for the Dirichlet Laplacian $-\Delta u=\lambda\|u\|_{L^q(\Omega)}^{2-q}|u|^{q-2}u$, $u\in H^1_0(\Omega)\setminus\{0\}$, whose first eigenvalue $\lambda_1(\Omega)$ is simple, with eigenfunctions of constant sign. The definition of a second eigenvalue is not unique. Following the approach introduced by Molle and Passaseo for the linear Laplacian, we construct the first non-trivial curve of the corresponding Fu\v{c}\'ik spectrum, and show that it has a unique diagonal point $(\gamma(\Omega),\gamma(\Omega))$. The Fu\v{c}\'ik-selected second value $\lambda_2^F(\Omega)=2^{1-\frac{2}{q}}\gamma(\Omega)$ is strictly larger than $\lambda_1(\Omega)$, admits a two-phase variational characterisation, and satisfies a sharp Hong-Krahn-Szeg\H{o} inequality: among sets of given measure it is minimised by the union of two equal balls. The selected solution at the diagonal point is a classical $(2,q)$-eigenfunction exactly when its nodal $L^q$-masses are balanced. We finally show that on balls in dimensions $N\geq2$ every Fu\v{c}\'ik-selected solution is foliated Schwarz symmetric.
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