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

TL;DR

This paper investigates a semilinear elliptic equation with an indefinite nonlinearity, proving existence and uniqueness of positive solutions near the linear regime, and analyzing spectral properties and inequalities related to the problem.

Contribution

It introduces a novel analysis of an indefinite nonlinear elliptic problem, establishing existence, uniqueness, nondegeneracy, and spectral properties in the asymptotically linear case.

Findings

Unique positive solutions exist for p close to 2

Spectral analysis reveals eigenfunction behavior and decay estimates

Analogues of classical inequalities are established in this setting

Abstract

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.