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

Mónica Clapp, Cristian Morales-Encinos, Alberto Saldaña, Mayra Soares

Abstract

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

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

Abstract

We study the semilinear elliptic problem where for a bounded smooth domain , , and . This equation arises in the study of optical waveguides and exhibits indefinite nonlinearity due to the sign-changing weight . We prove that, for sufficiently close to , the problem admits a unique positive solution, which is nondegenerate. Our approach combines a detailed analysis of an associated eigenvalue problem involving 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{ö} inequalities in this non-standard setting.

Paper Structure

This paper contains 13 sections, 26 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. The eigenvalue problem
  3. Existence of positive eigenvalues
  4. Properties of the eigenfunctions
  5. Monotonicity and scalings
  6. Decay
  7. Nodal domains
  8. Eigenfunctions, eigenvalues, and symmetries
  9. Nonexistence of negative eigenvalues
  10. The almost linear problem
  11. Asymptotic behavior of ground states
  12. Uniqueness of minimizers and nondegeneracy of positive solutions
  13. Sharp decay below and at the Serrin exponent

Key Result

Theorem 1.1

Let $\Omega$ be a bounded smooth open subset of $\mathbb{R}^N$ and $N\geq 3$. There exists $p_0>2$ such that the problem main has a unique positive least energy solution for $p\in (2,p_0)$. Moreover, all positive solutions of main are non-degenerate for $p\in (2,p_0)$.

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 43 more