Critical Ambrosetti-Prodi type problems on Carnot groups
Fuente:
arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917409701494784 |
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| author | Kanungo, Suman Mishra, Pawan Kumar |
| author_facet | Kanungo, Suman Mishra, Pawan Kumar |
| contents | In this paper, we investigate a class of critical Ambrosetti-Prodi type problems involving the sub-Laplacian on a Carnot group. Specifically, we consider
\[
\left\{
\begin{aligned}
-Δ_{\mathbb{G}} u &= λu + u_{+}^{2_{Q}^{*}-1} + f(ξ) \quad &&\text{in } Ω,\\[2mm]
u &= 0 \quad &&\text{on } \partialΩ,
\end{aligned}
\right.
\]
where $Δ_{\mathbb{G}}$ is the sub-Laplacian on a Carnot group $\mathbb{G}$, $Ω\subset \mathbb{G}$ is an open bounded domain with smooth boundary, $λ>0$ is a real parameter, $f\in L^{\infty}(Ω)$, $u_{+}$ denotes the positive part of $u$, and $2_{Q}^{*}$ is the critical Sobolev exponent associated with the homogeneous dimension $Q$. Motivated by the classical Ambrosetti-Prodi problem, we establish existence and multiplicity results for the cases $λ<λ_{1}$ and $λ>λ_{1}$, where $λ_{k}$ denotes the $k$-th Dirichlet eigenvalue of $-Δ_{\mathbb{G}}$. We also prove the existence of solutions at resonance when $λ=λ_{1}$ and show that bifurcation occurs from each eigenvalue $λ_{k}, k >1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_13591 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Critical Ambrosetti-Prodi type problems on Carnot groups Kanungo, Suman Mishra, Pawan Kumar Analysis of PDEs In this paper, we investigate a class of critical Ambrosetti-Prodi type problems involving the sub-Laplacian on a Carnot group. Specifically, we consider \[ \left\{ \begin{aligned} -Δ_{\mathbb{G}} u &= λu + u_{+}^{2_{Q}^{*}-1} + f(ξ) \quad &&\text{in } Ω,\\[2mm] u &= 0 \quad &&\text{on } \partialΩ, \end{aligned} \right. \] where $Δ_{\mathbb{G}}$ is the sub-Laplacian on a Carnot group $\mathbb{G}$, $Ω\subset \mathbb{G}$ is an open bounded domain with smooth boundary, $λ>0$ is a real parameter, $f\in L^{\infty}(Ω)$, $u_{+}$ denotes the positive part of $u$, and $2_{Q}^{*}$ is the critical Sobolev exponent associated with the homogeneous dimension $Q$. Motivated by the classical Ambrosetti-Prodi problem, we establish existence and multiplicity results for the cases $λ<λ_{1}$ and $λ>λ_{1}$, where $λ_{k}$ denotes the $k$-th Dirichlet eigenvalue of $-Δ_{\mathbb{G}}$. We also prove the existence of solutions at resonance when $λ=λ_{1}$ and show that bifurcation occurs from each eigenvalue $λ_{k}, k >1$. |
| title | Critical Ambrosetti-Prodi type problems on Carnot groups |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2604.13591 |