Liouville theorem for a class of p-Laplace type equations on manifolds

Abstract

We study a class of $p$-Laplace equations $${\Delta }_{p}u-\lambda u^{p-1}+u^{q-1}=0$$ on a closed $n$-dimensional Riemannian manifold $(M,g)$ with $\mathrm{Ric}⩾(n-1)g$. For $1<p<2$, $p<q<p^{\ast }$, and $0<\lambda <S_{p,q}^{-1}$, where $$S_{p,q}=\frac{q-p}{2}(\frac{p}{n}{)}^{\frac{p}{2}}(\frac{(p^{\ast }-1{)}^2(2-p)}{(p_{\ast }-1)(q-1)(p^{\ast }-q)}{)}^{\frac{2-p}{2}},$$ with $p_{\ast }=\frac{(n-1)p}{n-p}$ and $p^{\ast }=\frac{np}{n-p}$, we prove that the constant ${\lambda }^{\frac{1}{q-p}}$ is the unique positive solution of the equation. In contrast, for $p>2$ and $p<q<p^{\ast }$, the uniqueness fails for every $\lambda >0$; aside from the constant solution, the equation admits a positive nonconstant solution. This answers Véron's problem raised in \cite{Ver92}.