American Institute of Mathematical Sciences

May  2019, 39(5): 2731-2742. doi: 10.3934/dcds.2019114

Explicit estimates on positive supersolutions of nonlinear elliptic equations and applications

 1 School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran 2 Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, R3T 2N2, Canada

Received  May 2018 Revised  September 2018 Published  January 2019

Fund Project: The research of the second author was supported in part by NSERC

In this paper we consider positive supersolutions of the nonlinear elliptic equation
 $- \Delta u = \rho(x) f(u)|\nabla u|^p, ~~~~ {\rm{ in }}~~~~ \Omega,$
where
 $0\le p<1$
,
 $\Omega$
is an arbitrary domain (bounded or unbounded) in
 ${\mathbb{R}}^N$
(
 $N\ge 2$
),
 $f: [0, a_{f}) \rightarrow {\mathbb{R}}_{+}$
 $(0 < a_{f} \leq +\infty)$
is a non-decreasing continuous function and
 $\rho: \Omega \rightarrow \mathbb{R}$
is a positive function. Using the maximum principle we give explicit estimates on positive supersolutions
 $u$
at each point
 $x\in\Omega$
where
 $\nabla u\not\equiv0$
in a neighborhood of
 $x$
. As applications, we discuss the dead core set of supersolutions on bounded domains, and also obtain Liouville type results in unbounded domains
 $\Omega$
with the property that
 $\sup_{x\in\Omega}dist (x, \partial\Omega) = \infty$
. In particular when
 $\rho(x) = |x|^\beta$
(
 $\beta\in {\mathbb{R}}$
) and
 $f(u) = u^q$
with
 $q+p>1$
then every positive supersolution in an exterior domain is eventually constant if
 $(N-2)q+p(N-1)< N+\beta.$
Citation: Asadollah Aghajani, Craig Cowan. Explicit estimates on positive supersolutions of nonlinear elliptic equations and applications. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2731-2742. doi: 10.3934/dcds.2019114
References:

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