On the Fučík spectrum of the Logarithmic Laplacian
Rakesh Arora, Tuhina Mukherjee
TL;DR
This work analyzes the Fučík spectrum $Σ_L$ for the logarithmic Laplacian $L_Δ$ on bounded domains. By introducing a parametrized constrained energy $\tilde{E}_r$ and leveraging variational methods, it shows that the diagonal points $(λ_k^L,λ_k^L)$ belong to $Σ_L$ and that the vertical and horizontal lines through $λ_1^L$ are isolated in the spectrum. It then constructs the first nontrivial spectral curve, proving that its points take the form $(r+c(r), c(r))$ and $(c(r), r+c(r))$ with $r\ge 0$, and establishes detailed properties including Lipschitz continuity, monotonicity, and asymptotics. A variational characterization of the second eigenvalue $λ_2^L$ is obtained, and eigenfunctions for $λ>λ_1^L$ are shown to change sign. Finally, the paper addresses nonresonance issues relative to $Σ_L$ using mountain-pass techniques and Orlicz-type embeddings tailored to the logarithmic Laplacian framework.
Abstract
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 $λ_1^L$.