Existence of a bi-radial sign-changing solution for Hardy-Sobolev-Mazya type equation
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| author | Manna, Atanu Manna, Bhakti Bhusan |
| author_facet | Manna, Atanu Manna, Bhakti Bhusan |
| contents | In this article, we study the following Hardy-Sobolev-Maz'ya type equation:
\begin{equation}
-Δu - μ\frac{u}{|z|^2} = \frac{|u|^{q-2}u}{|z|^t}, \quad u \in D^{1,2} (\mathbb{R}^n),
\end{equation}
where $x = (y,z) \in \mathbb{R}^h \times \mathbb{R}^k = \mathbb{R}^n$, with $n \geq 5$, $2 < k <n$, and $t = n - \frac{(n-2)q}{2}$. We establish the existence of a bi-radial sign-changing solution under the assumptions $0 \leq μ< \frac{(k-2)^2}{4}, \, 2 < q <2^* = \frac{2(n-k+1)}{n-k-1}$. We approach the problem by lifting it to the hyperbolic setting, leading to the equation: $-Δ_{\mathbb{B}^N} u \, - \, λu = |u|^{p-1}u, \; u \in H^1(\mathbb{B}^N)$, $\mathbb{B}^N$ is the hyperbolic ball model. We study the existence of a sign-changing solution with suitable symmetry by constructing an appropriate invariant subspace of $H^1(\mathbb{B}^N)$ and applying the concentration compactness principle, and the corresponding solution of the Hardy-Sobolev-Maz'ya type equation becomes bi-radial under the corresponding isometry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_14224 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence of a bi-radial sign-changing solution for Hardy-Sobolev-Mazya type equation Manna, Atanu Manna, Bhakti Bhusan Analysis of PDEs In this article, we study the following Hardy-Sobolev-Maz'ya type equation: \begin{equation} -Δu - μ\frac{u}{|z|^2} = \frac{|u|^{q-2}u}{|z|^t}, \quad u \in D^{1,2} (\mathbb{R}^n), \end{equation} where $x = (y,z) \in \mathbb{R}^h \times \mathbb{R}^k = \mathbb{R}^n$, with $n \geq 5$, $2 < k <n$, and $t = n - \frac{(n-2)q}{2}$. We establish the existence of a bi-radial sign-changing solution under the assumptions $0 \leq μ< \frac{(k-2)^2}{4}, \, 2 < q <2^* = \frac{2(n-k+1)}{n-k-1}$. We approach the problem by lifting it to the hyperbolic setting, leading to the equation: $-Δ_{\mathbb{B}^N} u \, - \, λu = |u|^{p-1}u, \; u \in H^1(\mathbb{B}^N)$, $\mathbb{B}^N$ is the hyperbolic ball model. We study the existence of a sign-changing solution with suitable symmetry by constructing an appropriate invariant subspace of $H^1(\mathbb{B}^N)$ and applying the concentration compactness principle, and the corresponding solution of the Hardy-Sobolev-Maz'ya type equation becomes bi-radial under the corresponding isometry. |
| title | Existence of a bi-radial sign-changing solution for Hardy-Sobolev-Mazya type equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2505.14224 |