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

2025/11/07 by Mónica Clapp, Cristian Morales-Encinos, Clapp, Mónica +5
Computer Science · Mathematics · #35A02 (primary) 35J20 #35B09 35B40 (secondary) #35J61 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · doi:10.48550/arxiv.2511.05679

openalex publication_date 2025/11/07 · openalex created_date 2025/11/12 · openalex updated_date 2026/07/28

Abstract

We study the semilinear elliptic problem -Δu = Q_Ω |u|^p-2u in ℝ^N, where \( Q_Ω = χ_Ω - χ_{ℝ^N ∖ Ω} \) for a bounded smooth domain \( Ω⊂ ℝ^N \), \( N ≥ 3 \), and \( 1 < p < 2^* \). This equation arises in the study of optical waveguides and exhibits indefinite nonlinearity due to the sign-changing weight \( Q_Ω \). 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 \( Q_Ω \) 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--Szegö inequalities in this non-standard setting.

Citations

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• Asymptotic behavior of positive solutions of semilinear elliptic problems with increasing powers
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