Liouville type theorems for some $(p,q)$-Laplace equations with gradient dependent reaction on Riemannian manifolds

TL;DR

This work studies Liouville-type theorems and gradient estimates for the degenerate elliptic equation $\Delta_p u + \Delta_q u + h(u,|\nabla u|^2)=0$ on complete Riemannian manifolds, with $q\ge p>1$, using a blend of geometric and analytic tools. The authors derive gradient bounds under two structural assumptions on the gradient-reaction term $h$, employing Bochner formulas, Saloff-Coste's Sobolev inequality, and Nash–Moser iteration to control the nonlinear double-phase operator. A Liouville-type result is established: any nonnegative entire solution of $\Delta_p u + \Delta_q u = 0$ on a complete noncompact manifold with nonnegative Ricci curvature is constant when $n\le p \le q$, extending rigidity phenomena for $(p,q)$-harmonic functions. They also analyze the case with gradient-dependent reactions such as $h(u,|\nabla u|^2)=a|\nabla u|^r$, deriving explicit gradient estimates and consequent constant-solutions under suitable growth conditions, with applications to double-phase diffusion models in geometric contexts.

Abstract

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.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Corollary 1.5
• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3
• proof