Liouville properties for differential inequalities with $(p,q)$ Laplacian operator

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{$P_s$} -\Delta_p u-\Delta_q u\geq u^{s-1} \, \text{ in }\, \Omega, \end{equation*} where $1<q<p$, $s>1$ and $\Omega$ 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{$P_{sm}$} -\Delta_p u-\Delta_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 $\Omega=\mathbb{R}^N$ 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.