On the existence of global solutions of second-order quasilinear elliptic inequalities
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866916583948943360 |
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| author | Kon'kov, A. A. Shishkov, A. E. Surnachev, M. D. |
| author_facet | Kon'kov, A. A. Shishkov, A. E. Surnachev, M. D. |
| contents | We study the existence of global positive solutions of the differential inequalities $$
- \operatorname{div} A (x, u, \nabla u)
\ge
f (u)
\quad
\mbox{in } {\mathbb R}^n, $$ where $n \ge 2$ and $A$ is a Carathéodory function such that $$
(A (x, s, ζ) - A (x, s, ξ))(ζ- ξ) \ge 0, $$ $$
C_1
|ξ|^p
\le
ξ
A (x, s, ξ),
\quad
|A (x, s, ξ)|
\le
C_2
|ξ|^{p-1},
\quad
C_1, C_2 > 0,
\;
p > 1, $$ for almost all $x \in {\mathbb R}^n$ and for all $s \in {\mathbb R}$ and $ζ, ξ\in {\mathbb R}^n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_13076 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the existence of global solutions of second-order quasilinear elliptic inequalities Kon'kov, A. A. Shishkov, A. E. Surnachev, M. D. Analysis of PDEs 35A01, 35B44, 35B08, 35B09 We study the existence of global positive solutions of the differential inequalities $$ - \operatorname{div} A (x, u, \nabla u) \ge f (u) \quad \mbox{in } {\mathbb R}^n, $$ where $n \ge 2$ and $A$ is a Carathéodory function such that $$ (A (x, s, ζ) - A (x, s, ξ))(ζ- ξ) \ge 0, $$ $$ C_1 |ξ|^p \le ξ A (x, s, ξ), \quad |A (x, s, ξ)| \le C_2 |ξ|^{p-1}, \quad C_1, C_2 > 0, \; p > 1, $$ for almost all $x \in {\mathbb R}^n$ and for all $s \in {\mathbb R}$ and $ζ, ξ\in {\mathbb R}^n$. |
| title | On the existence of global solutions of second-order quasilinear elliptic inequalities |
| topic | Analysis of PDEs 35A01, 35B44, 35B08, 35B09 |
| url | https://arxiv.org/abs/2501.13076 |