Liouville type theorems for some $(p,q)$-Laplace equations with gradient dependent reaction on Riemannian manifolds
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866917183830884352 |
|---|---|
| author | Wang, Youde Zhang, Liqin |
| author_facet | Wang, Youde Zhang, Liqin |
| contents | In this paper, we combine Bochner formula, Saloff-Coste's Sobolev inequality and the Nash-Moser iteration method to study the local and global behaviors of solutions to the nonlinear elliptic equation $Δ_pu+Δ_qu+h(u,|\nabla u|^2)=0$ defined on a complete Riemannian manifold $\left(M,g\right)$, where $q\ge p>1$, $h\in C^1(\mathbb{R}\times\mathbb{R}^{+})$ and $Δ_z u=\mathrm{div}\left(\left|\nabla u\right|^{z-2}\nabla u\right)$, with $z\in\{ p,~q\}$, is the usual $z$-Laplace operator. Under some assumptions on $p$, $q$ and $h(x,y)$, we derive concise gradient estimates for solutions to the above equation and then obtain some Liouville type theorems. In particular, we use integral estimate method to show that, if $u$ is a non-negative entire solution to $Δ_p u +Δ_q u=0$ ($n\le p\le q$) on a complete non-compact Riemannian manifold $M$ with non-negative Ricci curvature and $\dim M = n\ge2$, then $u$ is a trivial constant solution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_01899 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Liouville type theorems for some $(p,q)$-Laplace equations with gradient dependent reaction on Riemannian manifolds Wang, Youde Zhang, Liqin Analysis of PDEs In this paper, we combine Bochner formula, Saloff-Coste's Sobolev inequality and the Nash-Moser iteration method to study the local and global behaviors of solutions to the nonlinear elliptic equation $Δ_pu+Δ_qu+h(u,|\nabla u|^2)=0$ defined on a complete Riemannian manifold $\left(M,g\right)$, where $q\ge p>1$, $h\in C^1(\mathbb{R}\times\mathbb{R}^{+})$ and $Δ_z u=\mathrm{div}\left(\left|\nabla u\right|^{z-2}\nabla u\right)$, with $z\in\{ p,~q\}$, is the usual $z$-Laplace operator. Under some assumptions on $p$, $q$ and $h(x,y)$, we derive concise gradient estimates for solutions to the above equation and then obtain some Liouville type theorems. In particular, we use integral estimate method to show that, if $u$ is a non-negative entire solution to $Δ_p u +Δ_q u=0$ ($n\le p\le q$) on a complete non-compact Riemannian manifold $M$ with non-negative Ricci curvature and $\dim M = n\ge2$, then $u$ is a trivial constant solution. |
| title | Liouville type theorems for some $(p,q)$-Laplace equations with gradient dependent reaction on Riemannian manifolds |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2601.01899 |