Some remarks on Liouville type results for quasilinear elliptic equations

Abstract

For a wide class of nonlinearities f ( u ) f(u) satisfying \[ f ( 0 ) = f ( a ) = 0 , f ( u ) > 0 in ( 0 , a ) and f ( u ) > 0 in ( a , ∞ ) , \mbox { $f(0)=f(a)=0$, $f(u)>0$ in $(0,a)$ and $f(u)>0$ in $(a,\infty )$,} \] we show that any nonnegative solution of the quasilinear equation − Δ p u = f ( u ) -\Delta _p u= f(u) over the entire R N \mathbb {R}^N must be a constant. Our results improve or complement some recently obtained Liouville type theorems. In particular, we completely answer a question left open by Du and Guo.