# 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:

show all references

##### References:
 [1] M. Á. Burgos-Pérez, J. García-Melián, A. Quaas. Classification of supersolutions and Liouville theorems for some nonlinear elliptic problems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4703-4721. doi: 10.3934/dcds.2016004 [2] Laura Baldelli, Roberta Filippucci. A priori estimates for elliptic problems via Liouville type theorems. Discrete & Continuous Dynamical Systems - S, 2020, 13 (7) : 1883-1898. doi: 10.3934/dcdss.2020148 [3] SYLWIA DUDEK, IWONA SKRZYPCZAK. Liouville theorems for elliptic problems in variable exponent spaces. Communications on Pure & Applied Analysis, 2017, 16 (2) : 513-532. doi: 10.3934/cpaa.2017026 [4] Alberto Farina. Symmetry of components, Liouville-type theorems and classification results for some nonlinear elliptic systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5869-5877. doi: 10.3934/dcds.2015.35.5869 [5] Pavol Quittner, Philippe Souplet. Parabolic Liouville-type theorems via their elliptic counterparts. Conference Publications, 2011, 2011 (Special) : 1206-1213. doi: 10.3934/proc.2011.2011.1206 [6] Phuong Le. Liouville theorems for stable weak solutions of elliptic problems involving Grushin operator. Communications on Pure & Applied Analysis, 2020, 19 (1) : 511-525. doi: 10.3934/cpaa.2020025 [7] Linfen Cao, Wenxiong Chen. Liouville type theorems for poly-harmonic Navier problems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 3937-3955. doi: 10.3934/dcds.2013.33.3937 [8] Philippe Souplet. Liouville-type theorems for elliptic Schrödinger systems associated with copositive matrices. Networks & Heterogeneous Media, 2012, 7 (4) : 967-988. doi: 10.3934/nhm.2012.7.967 [9] Xiaohui Yu. Liouville type theorem for nonlinear elliptic equation with general nonlinearity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4947-4966. doi: 10.3934/dcds.2014.34.4947 [10] Tomasz Adamowicz, Przemysław Górka. The Liouville theorems for elliptic equations with nonstandard growth. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2377-2392. doi: 10.3934/cpaa.2015.14.2377 [11] Quoc Hung Phan. Optimal Liouville-type theorems for a parabolic system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 399-409. doi: 10.3934/dcds.2015.35.399 [12] Phuong Le. Liouville theorems for an integral equation of Choquard type. Communications on Pure & Applied Analysis, 2020, 19 (2) : 771-783. doi: 10.3934/cpaa.2020036 [13] Xiaohui Yu. Liouville type theorems for singular integral equations and integral systems. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1825-1840. doi: 10.3934/cpaa.2016017 [14] Dong Li, Xinwei Yu. On some Liouville type theorems for the compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4719-4733. doi: 10.3934/dcds.2014.34.4719 [15] Chin-Chin Wu, Zhengce Zhang. Dead-core rates for the heat equation with a spatially dependent strong absorption. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2203-2210. doi: 10.3934/dcdsb.2013.18.2203 [16] Xinfu Chen, Jong-Shenq Guo, Bei Hu. Dead-core rates for the porous medium equation with a strong absorption. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1761-1774. doi: 10.3934/dcdsb.2012.17.1761 [17] Chunlai Mu, Jun Zhou, Yuhuan Li. Fast rate of dead core for fast diffusion equation with strong absorption. Communications on Pure & Applied Analysis, 2010, 9 (2) : 397-411. doi: 10.3934/cpaa.2010.9.397 [18] Italo Capuzzo Dolcetta, Antonio Vitolo. Glaeser's type gradient estimates for non-negative solutions of fully nonlinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 539-557. doi: 10.3934/dcds.2010.28.539 [19] Yutian Lei, Congming Li. Sharp criteria of Liouville type for some nonlinear systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3277-3315. doi: 10.3934/dcds.2016.36.3277 [20] Zhijun Zhang. Boundary blow-up for elliptic problems involving exponential nonlinearities with nonlinear gradient terms and singular weights. Communications on Pure & Applied Analysis, 2007, 6 (2) : 521-529. doi: 10.3934/cpaa.2007.6.521

2019 Impact Factor: 1.338