Abstract:In this paper, we apply a self-similar transformation to convert the parabolic equation with a Sobolev-Hardy term
\begin{align*} u_t-\Delta u= \frac{|u|^{q-2}u}{\left|x\right|^s} & \text { in } \mathbb{R}^N \times(0, \infty), \end{align*} into the following elliptic equation \begin{equation*} -\Delta v-\frac{1}{2} y \cdot \nabla v=\alpha v+ \frac{|v|^{q-2} v}{|y|^s}, \end{equation*} where $2 < q \leq 2^*(s)=\frac{2 N-2 s}{N-2}, 0 \leq s < 2, \alpha=\frac{2-s}{2q-4}$. For this equation, we establish the weighted Hardy inequality and Sobolev inequality. Furthermore, by virtue of the variational methods, we obtain infinitely many solutions in the subcritical case,
and prove the existence of solutions in the critical case. We also apply the Pohozaev identity to establish the nonexistence of solutions under certain conditions.
| Subjects: | Analysis of PDEs (math.AP) |
| Cite as: | arXiv:2605.24804 [math.AP] |
| (or arXiv:2605.24804v1 [math.AP] for this version) | |
| https://doi.org/10.48550/arXiv.2605.24804 arXiv-issued DOI via DataCite (pending registration) |
From: Fang Fei [view email]
[v1]
Sun, 24 May 2026 01:26:31 UTC (19 KB)
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