Liouville results for $(p,q)$-Laplacian elliptic equations with source terms involving gradient nonlinearities
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866912646123487232 |
|---|---|
| author | Bhakta, Mousomi Biswas, Anup Filippucci, Roberta |
| author_facet | Bhakta, Mousomi Biswas, Anup Filippucci, Roberta |
| contents | In this paper, we present a series of Liouville-type theorems for a class of nonhomogeneous quasilinear elliptic equations featuring reactions that depend on the solution and its gradient. Specifically, we investigate equations of the form $-Δ_p u - Δ_q u = f(u,\nabla u)$ with $p > q > 1$, where the nonlinearity $f$ takes forms such as $u^s|\nabla u|^m$ or $u^s + M|\nabla u|^m$ ($s, m\geq 0$). Our approach is twofold. For cases where the reaction term satisfies $|f(u,\nabla u)|\leq g(u)|\nabla u|^m$ with $m>q$ and $g$ is continuous, we prove that every bounded solution (without sign restriction) in $\mathbb{R}^N$ is constant by means of an Ishii-Lions type technique. In the remaining scenarios, we turn to the Bernstein method. The application of this method to the nonhomogeneous operator requires a nontrivial adaptation, as, roughly speaking, constant coefficients are replaced by functions that may not be bounded from above, which enables us to establish a crucial a priori estimate for the gradient of solutions in any domain $Ω$. This estimate, in turn, implies the desired Liouville properties on the entire space $\mathbb{R}^N$. As a consequence, we have fully extended Lions Liouville-type result for the Hamilton-Jacobi equation to the $(p,q)$-Laplacian setting, while for the $(p,q)$ generalized Lane-Emden equation, we provide an initial contribution in the direction of the classical result by Gidas and Spruck for $p=q=2$, as well as that of Serrin and Zou for $p=q$.
To the best of our knowledge, this is the first paper which studies Liouville properties for equations with nonhomogeneous operator involving source gradient terms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_12486 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Liouville results for $(p,q)$-Laplacian elliptic equations with source terms involving gradient nonlinearities Bhakta, Mousomi Biswas, Anup Filippucci, Roberta Analysis of PDEs 35J60, 35J92, 35J70, 35B08 In this paper, we present a series of Liouville-type theorems for a class of nonhomogeneous quasilinear elliptic equations featuring reactions that depend on the solution and its gradient. Specifically, we investigate equations of the form $-Δ_p u - Δ_q u = f(u,\nabla u)$ with $p > q > 1$, where the nonlinearity $f$ takes forms such as $u^s|\nabla u|^m$ or $u^s + M|\nabla u|^m$ ($s, m\geq 0$). Our approach is twofold. For cases where the reaction term satisfies $|f(u,\nabla u)|\leq g(u)|\nabla u|^m$ with $m>q$ and $g$ is continuous, we prove that every bounded solution (without sign restriction) in $\mathbb{R}^N$ is constant by means of an Ishii-Lions type technique. In the remaining scenarios, we turn to the Bernstein method. The application of this method to the nonhomogeneous operator requires a nontrivial adaptation, as, roughly speaking, constant coefficients are replaced by functions that may not be bounded from above, which enables us to establish a crucial a priori estimate for the gradient of solutions in any domain $Ω$. This estimate, in turn, implies the desired Liouville properties on the entire space $\mathbb{R}^N$. As a consequence, we have fully extended Lions Liouville-type result for the Hamilton-Jacobi equation to the $(p,q)$-Laplacian setting, while for the $(p,q)$ generalized Lane-Emden equation, we provide an initial contribution in the direction of the classical result by Gidas and Spruck for $p=q=2$, as well as that of Serrin and Zou for $p=q$. To the best of our knowledge, this is the first paper which studies Liouville properties for equations with nonhomogeneous operator involving source gradient terms. |
| title | Liouville results for $(p,q)$-Laplacian elliptic equations with source terms involving gradient nonlinearities |
| topic | Analysis of PDEs 35J60, 35J92, 35J70, 35B08 |
| url | https://arxiv.org/abs/2510.12486 |