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 uq(log u)ᵖ (q,p0) . 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 < m < 2 , m-1 < q < 1 , 0 < p < q ; (ii) m > 1 , q1 , 0 < p < 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.