On blow-up conditions for solutions of systems of quasilinear second-order elliptic inequalities
Fuente:
arXiv
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| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866908461361528832 |
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| author | Kon'kov, A. A. Shishkov, A. E. |
| author_facet | Kon'kov, A. A. Shishkov, A. E. |
| contents | We study systems of the differential inequalities $$
\left\{
\begin{aligned}
&
- \operatorname{div} A_1 (x, \nabla u_1)
\ge
F_1 (x, u_2)
&
\mbox{in } {\mathbb R}^n,
&
- \operatorname{div} A_2 (x, \nabla u_2)
\ge
F_2 (x, u_1)
&
\mbox{in } {\mathbb R}^n,
\end{aligned}
\right. $$ where $n \ge 2$ and $A_i$ are Caratheodory functions such that $$
C_1
|ξ|^{p_i}
\le
ξ
A_i (x, ξ),
\quad
|A_i (x, ξ)|
\le
C_2
|ξ|^{p_i - 1},
\quad i = 1,2, $$ with some constants $C_1, C_2 > 0$ and $p_1, p_2 > 1$ for almost all $x \in {\mathbb R}^n$ and for all $ξ\in {\mathbb R}^n$, $n \ge 2$. For non-negative solutions of these systems we obtain exact blow-up conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_04641 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On blow-up conditions for solutions of systems of quasilinear second-order elliptic inequalities Kon'kov, A. A. Shishkov, A. E. Analysis of PDEs 35B44, 35B08, 35J30, 35J70 We study systems of the differential inequalities $$ \left\{ \begin{aligned} & - \operatorname{div} A_1 (x, \nabla u_1) \ge F_1 (x, u_2) & \mbox{in } {\mathbb R}^n, & - \operatorname{div} A_2 (x, \nabla u_2) \ge F_2 (x, u_1) & \mbox{in } {\mathbb R}^n, \end{aligned} \right. $$ where $n \ge 2$ and $A_i$ are Caratheodory functions such that $$ C_1 |ξ|^{p_i} \le ξ A_i (x, ξ), \quad |A_i (x, ξ)| \le C_2 |ξ|^{p_i - 1}, \quad i = 1,2, $$ with some constants $C_1, C_2 > 0$ and $p_1, p_2 > 1$ for almost all $x \in {\mathbb R}^n$ and for all $ξ\in {\mathbb R}^n$, $n \ge 2$. For non-negative solutions of these systems we obtain exact blow-up conditions. |
| title | On blow-up conditions for solutions of systems of quasilinear second-order elliptic inequalities |
| topic | Analysis of PDEs 35B44, 35B08, 35J30, 35J70 |
| url | https://arxiv.org/abs/2404.04641 |