Existence and multiplicity of solutions for a critical Grushin problem with a singular nonlinearity
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915976462729216 |
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| author | Malhotra, Shammi |
| author_facet | Malhotra, Shammi |
| contents | We investigate the existence and multiplicity of positive solutions to the problem \begin{equation} \begin{cases} \begin{aligned}
- Δ_γ u &= λu^{p} + u^{-δ} &\quad \text{in } Ω,
\quad u &= 0 &\quad \text{on } \partial Ω, \end{aligned} \end{cases} \end{equation} where $Δ_γ$ denotes the Grushin operator defined by \begin{equation} Δ_γ := Δ_x + (1+γ)^2 |x|^{2γ}Δ_y, \end{equation} with $γ>0$, $z=(x,y)\in \mathbb{R}^N$, $N=n+m$, $n \geq 1$, $m\geq 1$, $Ω\subset \mathbb{R}^N$ a smooth bounded domain, $λ>0$, $1<p<\infty$, and $δ>0$.
The analysis depends on the exponent $p$, which may be subcritical, critical, or supercritical, that is, $p<2_γ^*-1$, $p=2_γ^*-1$, or $p>2_γ^*-1$, respectively, where $2_γ^*=\frac{2Q}{Q-2}$ is the critical Sobolev exponent associated with the Grushin operator, and $Q=m+(1+γ)n$ is the corresponding homogeneous dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_02020 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Existence and multiplicity of solutions for a critical Grushin problem with a singular nonlinearity Malhotra, Shammi Analysis of PDEs We investigate the existence and multiplicity of positive solutions to the problem \begin{equation} \begin{cases} \begin{aligned} - Δ_γ u &= λu^{p} + u^{-δ} &\quad \text{in } Ω, \quad u &= 0 &\quad \text{on } \partial Ω, \end{aligned} \end{cases} \end{equation} where $Δ_γ$ denotes the Grushin operator defined by \begin{equation} Δ_γ := Δ_x + (1+γ)^2 |x|^{2γ}Δ_y, \end{equation} with $γ>0$, $z=(x,y)\in \mathbb{R}^N$, $N=n+m$, $n \geq 1$, $m\geq 1$, $Ω\subset \mathbb{R}^N$ a smooth bounded domain, $λ>0$, $1<p<\infty$, and $δ>0$. The analysis depends on the exponent $p$, which may be subcritical, critical, or supercritical, that is, $p<2_γ^*-1$, $p=2_γ^*-1$, or $p>2_γ^*-1$, respectively, where $2_γ^*=\frac{2Q}{Q-2}$ is the critical Sobolev exponent associated with the Grushin operator, and $Q=m+(1+γ)n$ is the corresponding homogeneous dimension. |
| title | Existence and multiplicity of solutions for a critical Grushin problem with a singular nonlinearity |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2605.02020 |