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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2505.11768 |
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| _version_ | 1866911106541289472 |
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| author | Lopera, Emer Recôva, Leandro Rumbos, Adolfo |
| author_facet | Lopera, Emer Recôva, Leandro Rumbos, Adolfo |
| contents | In this paper, we study a class of problems proposed by Servadei and Valdinoci in \cite{Ser3}; namely, \begin{equation}\label{prob_0} \left\{\begin{aligned} -\mathcal{L}_{K} u(x)-λu(x) & =f(x,u), \mbox{ for } x\in Ω; u & =0 \quad \text{ in } \mathbb{R}^{N}\backslashΩ, \end{aligned} \right. \end{equation} where $Ω\subset \mathbb{R}^{N}$ is an open bounded set with Lipschitz boundary, $λ\in\mathbb{R}$, $f\in C^{1}(\overlineΩ\times\mathbb{R},\mathbb{R})$, with $f(x,0) = 0$ for $x\inΩ$, and $\mathcal{L}_K$ is a non-local integrodifferential operator with homogeneous Dirichlet boundary condition. By computing the critical groups of the associated energy functional for problem (1) at the origin and at infinity, respectively, we prove that problem (\ref{prob_0}) has three nontrivial solutions for the case $λ< λ_1$ and two nontrivial solutions for the case $λ\geqslantλ_1,$ where $λ_1$ is the first eigenvalue of the operator $-\mathcal{L}_K$. Finally, assuming that the nonlinearity $f$ is odd in the second variable, we prove the existence of an unbounded sequence of weak solutions of problem (1) for the case $λ\geqslantλ_1$. We use variational methods and infinite-dimensional Morse theory to obtain the results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_11768 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multiplicity results for non-local operators of elliptic type Lopera, Emer Recôva, Leandro Rumbos, Adolfo Analysis of PDEs 35J20 In this paper, we study a class of problems proposed by Servadei and Valdinoci in \cite{Ser3}; namely, \begin{equation}\label{prob_0} \left\{\begin{aligned} -\mathcal{L}_{K} u(x)-λu(x) & =f(x,u), \mbox{ for } x\in Ω; u & =0 \quad \text{ in } \mathbb{R}^{N}\backslashΩ, \end{aligned} \right. \end{equation} where $Ω\subset \mathbb{R}^{N}$ is an open bounded set with Lipschitz boundary, $λ\in\mathbb{R}$, $f\in C^{1}(\overlineΩ\times\mathbb{R},\mathbb{R})$, with $f(x,0) = 0$ for $x\inΩ$, and $\mathcal{L}_K$ is a non-local integrodifferential operator with homogeneous Dirichlet boundary condition. By computing the critical groups of the associated energy functional for problem (1) at the origin and at infinity, respectively, we prove that problem (\ref{prob_0}) has three nontrivial solutions for the case $λ< λ_1$ and two nontrivial solutions for the case $λ\geqslantλ_1,$ where $λ_1$ is the first eigenvalue of the operator $-\mathcal{L}_K$. Finally, assuming that the nonlinearity $f$ is odd in the second variable, we prove the existence of an unbounded sequence of weak solutions of problem (1) for the case $λ\geqslantλ_1$. We use variational methods and infinite-dimensional Morse theory to obtain the results. |
| title | Multiplicity results for non-local operators of elliptic type |
| topic | Analysis of PDEs 35J20 |
| url | https://arxiv.org/abs/2505.11768 |