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

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 $\lambda_1^L$.