Liouville theorems for $p$-Laplacian equations in convex cones without finite-energy condition

Abstract

We study the anisotropic Finsler $p$-Laplacian equation \begin{equation*} \left\{ \begin{aligned} &-Δ^{H}_{p}u=f(u) \quad\,\,\, &{\rm{in}} \,\, \mathcal{C}, &{\bf{a}}(\nabla u)\cdot ν=0 \quad\,\,\, &{\rm{on}} \,\, \partial\mathcal{C}, \end{aligned} \right. \end{equation*} where $N\ge 3$, $1<p<N$, $\mathcal{C}\subseteq {\mathbb{R}}^N$ is an open convex cone and ${\Delta }_p^H$ is the anisotropic Finsler $p$-Laplacian operator. If $f(u)$ is nonnegative and subcritical, we prove that every bounded nonnegative solution in $\mathcal{C}$ is identically zero. In particular, for $f(u)=u^q$ with $0<q<p^{\ast }-1$, we establish a pointwise decay estimate in $\mathcal{C}$ via the doubling argument and blowing-up method and prove that all nonnegative solutions must be zero without the boundedness assumption. Our results are the subcritical counterpart of the classification result for the critical case in \cite{CFR}, and extend the Liouville type theorems in ${\mathbb{R}}^N$ for the standard $p$-Laplacian in \cite{SZ} and for the anisotropic $p$-Laplacian in \cite{CFV, CHN} to general convex cones $\mathcal{C}$. In the critical case $f(u)=u^{p^{\ast }-1}$ and typical case $H(\xi )=|\xi |$, for $\frac{N+1}{3}<p<N$, we classified the positive solutions of the critical $p$-Laplacian equation in convex cones $\mathcal{C}$ without finite-energy assumption. This extends the classification result of \cite{Ou} in ${\mathbb{R}}^N$ to general convex cones $\mathcal{C}$, and removes the finite-energy assumption in \cite{CFR} in the typical case $H(\xi )=|\xi |$.