Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2601.03865 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866915713891958784 |
|---|---|
| author | Arora, Rakesh Mukherjee, Tuhina |
| author_facet | Arora, Rakesh Mukherjee, Tuhina |
| contents | 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 $λ_1^L$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_03865 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Fučík spectrum of the Logarithmic Laplacian Arora, Rakesh Mukherjee, Tuhina Analysis of PDEs 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 $λ_1^L$. |
| title | On the Fučík spectrum of the Logarithmic Laplacian |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2601.03865 |