Liouville-type theorems for the quasilinear elliptic equation and inequality with logarithmic nonlinearity

This paper is concerned with Liouville-type theorems of positive weak solutions to elliptic $m$ -Laplace equation and inequality with the logarithmic nonlinearity $u^q(\log u)^p (q,p\geqslant0)$ . Using a direct Bernstein method we obtain a first range of values of $m,q,p$ in which all positive weak solutions of equation are constants, this holds in the following cases: (i) $1 \lt m \lt 2$ , $m-1 \lt q \lt 1$ , $0 \lt p \lt q$ ; (ii) $m \gt 1$ , $q\geqslant1$ , $0 \lt p \lt 1$ . When $q=1$ , the positive weak solutions are required to be bounded. Based on transformation of inequality and the utilization of suitable cut-off functions, we establish a Liouville-type theorem for positive weak solutions of inequality; this result also remains valid on complete noncompact Riemannian manifold.