Some remarks on Liouville type results for quasilinear elliptic equations

Dancer, E.; Du, Yihong

doi:10.1090/S0002-9939-02-06733-3

Some remarks on Liouville type results for quasilinear elliptic equations
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by E. N. Dancer and Yihong Du PDF

Proc. Amer. Math. Soc. 131 (2003), 1891-1899 Request permission

Abstract:

For a wide class of nonlinearities $f(u)$ satisfying \[ \mbox { $f(0)=f(a)=0$, $f(u)>0$ in $(0,a)$ and $f(u)<0$ in $(a,\infty )$,}\] we show that any nonnegative solution of the quasilinear equation $-\Delta _p u= f(u)$ over the entire $\mathbb {R}^N$ 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.

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