Jinzi Bai, Fei Fang
Abstract:
In this article, we study the Fucik spectrum for operators with rapidly increasing
weight, which is defined as a set \(\Sigma\) comprising those
\((\alpha, \beta) \in \mathbb{R}^2\) such that
$$\displaylines{
L u:=-\Delta u-\frac{1}{2}(x \cdot \nabla u)=\alpha u^{+}-\beta u^{-}, \quad\text{in }
\mathbb{R}^N,\cr
u\in X,
}$$
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 Fucik problem on bounded
domains. As an application, we establish the
multiplicity of solutions to the problem
$$\displaylines{
-\Delta u-\frac{1}{2}(x \cdot \nabla u)=f(x,u), \quad\text{in } \mathbb{R}^N,\cr
u\in X,
}$$
where, \(N\geq1\) and the nonlinearity \(f\) is asymptotically linear at zero and at
infinity.
Submitted May 1, 2026. Published July 29, 2026.
Math Subject Classifications: 35R11, 35R09, 35A15.
Key Words: Fucik spectrum; self-similar problem; multiplicity of solutions.
DOI: 10.58997/ejde.2026.58
Show me the PDF file (441 KB), TEX file for this article.
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Jinzi Bai School of Mathematics and Statistics Beijing Technology and Business University Beijing 100048, China email: 1621026802@qq.com |
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Fei Fang School of Mathematics and Statistics Beijing Technology and Business University Beijing 100048, China email: fangfei68@163.com |
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