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On the Fučík spectrum of the Logarithmic Laplacian

In this paper, we investigate the Fučík spectrum $Σ_L$ associated with the logarithmic Laplacian. This spectrum is defined as the set of all pairs $(α,β) \in \mathbb{R}^2$ for which the problem \[ L_Δu = αu^+-βu^- ~\text{in} ~ Ω\quad \text{and} \quad u=0 ~\text{in} ~\mathbb{R}^N\setminus Ω\] admits a nontrivial solution $u$. Here, $Ω\subset \mathbb{R}^N$ is a bounded domain with $C^{1,1}$ boundary, $u^\pm = \max\{\pm u,0\}$, and $u = u^+ - u^-$. We show that the lines $λ_1^L \times \mathbb{R}$ and $\mathbb{R} \times λ_1^L$, where $λ_1^L$ denotes the first eigenvalue of $L_Δ$, lies in the spectrum $Σ_L$ and are isolated within the spectrum. Furthermore, we establish the existence of the first nontrivial curve in $Σ_L$ 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 $λ> λ_1^L$ are sign-changing. Finally, we address a nonresonance problem with respect to the Fučík spectrum $Σ_L$, employing variational methods and carefully overcoming the difficulties arising from the contrasting features of the first eigenvalue $λ_1^L$.