Liouville properties for differential inequalities with $(p,q)$ Laplacian operator
Mousomi Bhakta, Anup Biswas, Roberta Filippucci
TL;DR
This work establishes Liouville-type nonexistence and rigidity results for differential inequalities driven by the nonhomogeneous $(p,q)$-Laplacian. It analyzes two core problems: a power-type nonlinearity $-\,\Delta_p u-\,\Delta_q u\ge u^{s-1}$ in exterior domains and a gradient nonlinearity $-\,\Delta_p u-\,\Delta_q u\ge u^s|\nabla u|^m$ in $\mathbb{R}^N$, deriving sharp thresholds around the Serrin exponent $q_* = \frac{q(N-1)}{N-q}$ and constructing new lower-bound iteration techniques that complement or replace capacity-based methods. The results show sharp nonexistence for subcritical regimes, existence or triviality at critical values, and constancy of positive solutions under various subcritical gradient-exponent ranges, significantly extending classical results from the homogeneous $p$-Laplacian to the $(p,q)$-Laplacian. The methods deliver a robust framework for handling nonhomogeneous operators with unbalanced growth, with potential implications for reaction-diffusion systems and related nonlinear PDEs.
Abstract
In this paper, we establish several Liouville-type theorems for a class of nonhomogenenous quasilinear inequalities. In the first part, we prove various Liouville results associated with nonnegative solutions to \begin{equation*}\tag{$P_s$} -Δ_p u-Δ_q u\geq u^{s-1} \, \text{ in }\, Ω, \end{equation*} where $1<q<p$, $s>1$ and $Ω$ is any exterior domain of $\mathbb{R}^N$. In particular, we prove that for $q<N$, inequality $(P_s)$ does not admit any positive solution when $s<q_*$ and $(P_s)$ admits a positive solution if $s>q_*$, where $q_*=\frac{q(N-1)}{N-q}$ is the Serrin exponent for the $q$-Laplacian. Further, we show that when $s=q_*$ and $p<s$ the only nonnegative solution to $(P_s)$ is the trivial solution. On the other hand, for $q\geq N$ we prove that $u\equiv 0$ is the only nonnegative solution for $(P_s)$ for any $s>1$. In the second part, we consider the inequality \begin{equation*}\tag{$P_{sm}$} -Δ_p u-Δ_q u \geq u^s |\nabla u|^m \quad \text{ in }\mathbb{R}^N, \end{equation*} where $1<q<p$, $N>q$ and $s, \, m\geq 0$. We prove that, for $\{0\leq m\leq q-1\}\cup\{m>p-1\}$, the only positive solution to $(P_{sm})$ is constant, provided $s(N-q)+m(N-1)<N(q-1)$. This, in particular, proves that if $Ω=\mathbb{R}^N$ then any nonnegative solution to $(P_s)$ with $1<q<N$ and $1<s<q_*$ is the trivial solution. To prove Liouville in the range $0\leq m<q-1$, we first prove an almost optimal lower estimate of any nonnegative supersolution of $(P_{sm})$ and then leveraging this estimate we prove Liouville result. To the best of our knowledge, this technique is completely new and provides an alternative approach to the capacity method of Mitidieri-Pohozaev provided higher regularity is available.