On the Fučík spectrum of the Logarithmic Laplacian

Abstract

In this paper, we investigate the Fučík 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 \[ L_Δu = αu^+-βu^- ~\text{in} ~ Ω\quad \text{and} \quad u=0 ~\text{in} ~\mathbb{R}^N\setminus Ω\] admits a nontrivial solution $u$. Here, $\Omega \subset {\mathbb{R}}^N$ 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 Fučík spectrum ${\Sigma }_L$, employing variational methods and carefully overcoming the difficulties arising from the contrasting features of the first eigenvalue ${\lambda }_1^L$.

21 pages. Comments are welcome