On the Fu\v{c}\'{i}k spectrum of the Logarithmic Laplacian

TL;DR

This paper studies the Fučík spectrum of the logarithmic Laplacian, identifying key spectral lines, establishing the existence and properties of the first nontrivial curve, and analyzing eigenvalues and eigenfunctions.

Contribution

It introduces the first nontrivial curve in the Fučík spectrum of the logarithmic Laplacian and provides a variational characterization of the second eigenvalue.

Findings

Lines $\lambda_1^L imes \mathbb{R}$ and $\mathbb{R} imes \lambda_1^L$ are in the spectrum and isolated.

Existence and qualitative properties of the first nontrivial curve are established.

Eigenfunctions for eigenvalues greater than $\lambda_1^L$ are sign-changing.

Abstract

In this paper, we investigate the Fu\v{c}\'{i}k spectrum $\Sigma_L$ associated with the logarithmic Laplacian. This spectrum is defined as the set of all pairs $(\alpha,\beta) \in \mathbb{R}^2$ for which the problem \[ L_\Delta u = \alpha u^+-\beta u^- ~\text{in} ~ \Omega \quad \text{and} \quad u=0 ~\text{in} ~\mathbb{R}^N\setminus \Omega \] admits a nontrivial solution $u$. Here, $\Omega \subset \mathbb{R}^N$ is a bounded domain with $C^{1,1}$ boundary, $u^\pm = \max\{\pm u,0\}$, and $u = u^+ - u^-$. We show that the lines $\lambda_1^L \times \mathbb{R}$ and $\mathbb{R} \times \lambda_1^L$, where $\lambda_1^L$ denotes the first eigenvalue of $L_\Delta$, lies in the spectrum $\Sigma_L$ and are isolated within the spectrum. Furthermore, we establish the existence of the first nontrivial curve in $\Sigma_L$ and analyze its qualitative properties, including Lipschitz continuity, strict monotonicity, and asymptotic behavior. In addition, we obtain a variational characterization of the second eigenvalue of the logarithmic Laplacian and show that all eigenfunctions corresponding to eigenvalues $\lambda > \lambda_1^L$ are sign-changing. Finally, we address a nonresonance problem with respect to the Fu\v{c}\'{i}k spectrum $\Sigma_L$, employing variational methods and carefully overcoming the difficulties arising from the contrasting features of the first eigenvalue $\lambda_1^L$.