Liouville results for $(p,q)$-Laplacian elliptic equations with source terms involving gradient nonlinearities

TL;DR

The paper establishes Liouville-type results for the nonhomogeneous $(p,q)$-Laplacian $-\\Delta_p u-\\Delta_q u=f(u,\\nabla u)$ with $p>q>1$, where the source exhibits gradient nonlinearities. It develops two complementary approaches: an Ishii–Lions technique for gradient-dominated reactions and a Bernstein-type method adapted to the nonhomogeneous operator to obtain sharp gradient estimates that imply Liouville properties on $\\mathbb{R}^N$. The results extend Lions’ Liouville theorem to the $(p,q)$-Laplacian in the Hamilton–Jacobi regime and provide initial contributions toward Gidas–Spruck/Serrin–Zou-type results for the generalized Lane–Emden equation in this nonhomogeneous setting. The work also analyzes sums of nonlinearities and demonstrates regime-dependent dominance of the $p$-Laplacian, yielding a priori gradient bounds and global constancy results under broad parameter ranges. Overall, the paper broadens Liouville theory to nonhomogeneous operators with gradient source terms, combining viscosity/IBP techniques with carefully tailored a priori estimates.

Abstract

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.

Theorems & Definitions (17)

• Definition 1
• Theorem 1
• Remark 1
• Theorem 2
• Remark 2
• Theorem 3
• Remark 3
• Theorem 4
• Theorem 5
• Remark 4