Fučik spectrum for the operator with rapidly increasing weight and applications
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| Formato: | Preprint |
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2026
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| _version_ | 1866908978044207104 |
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| author | Bai, Jinzi Fang, Fei |
| author_facet | Bai, Jinzi Fang, Fei |
| contents | In this paper, we study the Fučik spectrum for the operator with rapidly increasing weight, which is defined as a set $Σ$ comprising those $(α, β) \in \mathbb{R}^2$ such that \begin{equation*} \left\{\begin{array}{l} L u:=-Δu-\frac{1}{2}(x \cdot \nabla u)=αu^{+}-βu^{-}, \text{in}\ \mathbb{R}^N,\\ u\in X, \end{array}\right. \end{equation*} has a non-trivial solution $u$, where, $N\geq1$, $u^{ \pm}=\max ( \pm u, 0)$, $u=u^{+}-u^{-}$. The existence of a first nontrivial curve $\mathcal{C}$
of this spectrum, along with some of its properties (e.g., Lipschitz continuity, strict decrease and asymptotic behavior) is investigated in this paper. Our difficulty is that the problem is defined on the whole space $\mathbb{R}^N$, and therefore certain estimates do not carry over from the Fučik problem on bounded domains. As an application, we establish the multiplicity of solutions to the following problem \begin{equation*} \left\{\begin{array}{l} -Δu-\frac{1}{2}(x \cdot \nabla u)=f(x,u), \text{in}\ \mathbb{R}^N,\\ u\in X, \end{array}\right. \end{equation*} where, $N\geq1$ and the nonlinearity $f$ is asymptotically linear at zero and at infinity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_17280 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fučik spectrum for the operator with rapidly increasing weight and applications Bai, Jinzi Fang, Fei Analysis of PDEs 35R11, 35R09, 35A15 In this paper, we study the Fučik spectrum for the operator with rapidly increasing weight, which is defined as a set $Σ$ comprising those $(α, β) \in \mathbb{R}^2$ such that \begin{equation*} \left\{\begin{array}{l} L u:=-Δu-\frac{1}{2}(x \cdot \nabla u)=αu^{+}-βu^{-}, \text{in}\ \mathbb{R}^N,\\ u\in X, \end{array}\right. \end{equation*} has a non-trivial solution $u$, where, $N\geq1$, $u^{ \pm}=\max ( \pm u, 0)$, $u=u^{+}-u^{-}$. The existence of a first nontrivial curve $\mathcal{C}$ of this spectrum, along with some of its properties (e.g., Lipschitz continuity, strict decrease and asymptotic behavior) is investigated in this paper. Our difficulty is that the problem is defined on the whole space $\mathbb{R}^N$, and therefore certain estimates do not carry over from the Fučik problem on bounded domains. As an application, we establish the multiplicity of solutions to the following problem \begin{equation*} \left\{\begin{array}{l} -Δu-\frac{1}{2}(x \cdot \nabla u)=f(x,u), \text{in}\ \mathbb{R}^N,\\ u\in X, \end{array}\right. \end{equation*} where, $N\geq1$ and the nonlinearity $f$ is asymptotically linear at zero and at infinity. |
| title | Fučik spectrum for the operator with rapidly increasing weight and applications |
| topic | Analysis of PDEs 35R11, 35R09, 35A15 |
| url | https://arxiv.org/abs/2604.17280 |