Multiple sign-changing solutions for semilinear subelliptic Dirichlet problem
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914087674314752 |
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| author | Chen, Hua Chen, Hong-Ge Li, Jin-Ning Liao, Xin |
| author_facet | Chen, Hua Chen, Hong-Ge Li, Jin-Ning Liao, Xin |
| contents | We study the following perturbation from symmetry problem for the semilinear subelliptic equation \[ \left\{
\begin{array}{cc}
-\triangle_{X} u=f(x,u)+g(x,u) & \mbox{in}~Ω, \\[2mm]
u\in H_{X,0}^{1}(Ω),\hfill
\end{array}
\right. \] where $\triangle_{X}=-\sum_{i=1}^{m}X_{i}^{*}X_{i}$ is the self-adjoint sub-elliptic operator associated with Hörmander vector fields $X=(X_{1},X_{2},\ldots,X_{m})$, $Ω$ is an open bounded subset in $\mathbb{R}^n$, and $H_{X,0}^{1}(Ω)$ denotes the weighted Sobolev space. We establish multiplicity results for sign-changing solutions using a perturbation method alongside refined techniques for invariant sets. The pivotal aspect lies in the estimation of the lower bounds of min-max values associated with sign-changing critical points. In this paper, we construct two distinct lower bounds of these min-max values. The first one is derived from the lower bound of Dirichlet eigenvalues of $-\triangle_{X}$, while the second one is based on the Morse-type estimates and Cwikel-Lieb-Rozenblum type inequality in degenerate cases. These lower bounds provide different sufficient conditions for multiplicity results, each with unique advantages and are not mutually inclusive, particularly in the general non-equiregular case. This novel observation suggests that in some sense, the situation for sub-elliptic equations would have essential difference from the classical elliptic framework. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_10120 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multiple sign-changing solutions for semilinear subelliptic Dirichlet problem Chen, Hua Chen, Hong-Ge Li, Jin-Ning Liao, Xin Analysis of PDEs 35A15, 35H20, 35J70 We study the following perturbation from symmetry problem for the semilinear subelliptic equation \[ \left\{ \begin{array}{cc} -\triangle_{X} u=f(x,u)+g(x,u) & \mbox{in}~Ω, \\[2mm] u\in H_{X,0}^{1}(Ω),\hfill \end{array} \right. \] where $\triangle_{X}=-\sum_{i=1}^{m}X_{i}^{*}X_{i}$ is the self-adjoint sub-elliptic operator associated with Hörmander vector fields $X=(X_{1},X_{2},\ldots,X_{m})$, $Ω$ is an open bounded subset in $\mathbb{R}^n$, and $H_{X,0}^{1}(Ω)$ denotes the weighted Sobolev space. We establish multiplicity results for sign-changing solutions using a perturbation method alongside refined techniques for invariant sets. The pivotal aspect lies in the estimation of the lower bounds of min-max values associated with sign-changing critical points. In this paper, we construct two distinct lower bounds of these min-max values. The first one is derived from the lower bound of Dirichlet eigenvalues of $-\triangle_{X}$, while the second one is based on the Morse-type estimates and Cwikel-Lieb-Rozenblum type inequality in degenerate cases. These lower bounds provide different sufficient conditions for multiplicity results, each with unique advantages and are not mutually inclusive, particularly in the general non-equiregular case. This novel observation suggests that in some sense, the situation for sub-elliptic equations would have essential difference from the classical elliptic framework. |
| title | Multiple sign-changing solutions for semilinear subelliptic Dirichlet problem |
| topic | Analysis of PDEs 35A15, 35H20, 35J70 |
| url | https://arxiv.org/abs/2510.10120 |