January/February 2023 Nonlinear Liouville-type theorems for generalized Baouendi-Grushin operator on Riemannian manifolds
Mohamed Jleli, Maria Alessandra Ragusa, Bessem Samet
Adv. Differential Equations 28(1/2): 143-168 (January/February 2023). DOI: 10.57262/ade028-0102-143


We are concerned with differential inequalities of the form $$ -[d_{M_2}(y_0,y)]^{2\rho_2}\Delta_{M_1} u-[d_{M_1}(x_0,x)]^{2\rho_1}\Delta_{M_2}u\geq V |u|^p\mbox { in } M_1\times M_2, $$ where $M_i$ ($i=1,2$) are complete noncompact Riemannian manifolds, $(x_0,y_0)\in M_1\times M_2$ is fixed, $d_{M_1}(x_0,\cdot)$ is the distance function on $M_1$, $d_{M_2}(y_0,\cdot)$ is the distance function on $M_2$, $\Delta_{M_i}$ is the Laplace-Beltrami operator on $M_i$, $V=V(x,y) > 0$ is a measurable function, and $p>1$. Namely, we establish necessary conditions for existence of nontrivial weak solutions to the considered problem. The obtained conditions depend on the parameters of the problem as well as the geometry of the manifolds $M_i$. Next, we discuss some special cases of potential functions $V$. The proof of our main result is based on the nonlinear capacity method and a result due to Bianchi and Setti (2018) about the construction of cut-off functions with controlled gradient and Laplacian, under certain assumptions on the Ricci curvatures of the manifolds.


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Mohamed Jleli. Maria Alessandra Ragusa. Bessem Samet. "Nonlinear Liouville-type theorems for generalized Baouendi-Grushin operator on Riemannian manifolds." Adv. Differential Equations 28 (1/2) 143 - 168, January/February 2023. https://doi.org/10.57262/ade028-0102-143


Published: January/February 2023
First available in Project Euclid: 12 September 2022

Digital Object Identifier: 10.57262/ade028-0102-143

Primary: 35B33 , 35B53 , 35R01

Rights: Copyright © 2023 Khayyam Publishing, Inc.


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Vol.28 • No. 1/2 • January/February 2023
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