On the asymptotically linear problem for an elliptic equation with an indefinite nonlinearity

We study the semilinear elliptic problem $$-\Delta u = Q_{\Omega} |u|^{p-2}u \quad \text{in } \mathbb{R}^N,$$ where $Q_{\Omega} = \chi_{\Omega} - \chi_{\mathbb{R}^N \setminus \Omega}$ for a bounded smooth domain $\Omega \subset \mathbb{R}^N$, $N \ge 3$, and $1 < p < 2^{*}$. This equation arises in the study of optical waveguides and exhibits indefinite nonlinearity due to the sign-changing weight $Q_{\Omega}$. We prove that, for $p > 2$ sufficiently close to $2$, the problem admits a unique positive solution, which is nondegenerate. Our approach combines a detailed analysis of an associated eigenvalue problem involving $Q_{\Omega}$ with variational methods and blow-up techniques in the asymptotically linear regime. We also provide a comprehensive study of the spectral properties of the corresponding linear problem, including the existence and qualitative behavior of eigenfunctions, sharp decay estimates, and symmetry results. In particular, we establish analogues of the Faber--Krahn and Hong--Krahn--Szeg{\"o} inequalities in this non-standard setting.