Publication detail

Positive supersolutions of non-autonomous quasilinear elliptic equations with mixed reaction

AGHAJANI, A. RADULESCU, V.

Original Title

Positive supersolutions of non-autonomous quasilinear elliptic equations with mixed reaction

Type

journal article in Web of Science

Language

English

Original Abstract

We provide a simple method for obtaining new Liouville-type theorems for positive supersolutions of the We We provide a simple method for obtaining new Liouville-type theorems for positive supersolutions of the elliptic problem - Delta(p)u+ b(x)vertical bar del u vertical bar(pq/q+1) = c(x)u(q) in Omega, where Omega is an exterior domain in R-N with N >= p > 1 and q >= p - 1. In the case q not equal p - 1, we mainly deal with potentials of the type b(x) = vertical bar x vertical bar(a), c(x) = lambda vertical bar x vertical bar(sigma), where lambda > 0 and a, sigma is an element of R. We show that positive supersolutions do not exist in some ranges of the parameters p, q, a, sigma, which turn out to be optimal. When q = p - 1, we consider the above problem with general weights b(x) >= 0, c(x) > 0 and we assume that c(x)- b(p)(x)/p(p) > 0 for large vertical bar x vertical bar, but we also allow the case lim(vertical bar x vertical bar ->infinity)[c(x)- b(p)(x)/p(p)] = 0. The weights b and c are allowed to be unbounded. We prove that if this equation has a positive supersolution, then the potentials must satisfy a related differential inequality not depending on the supersolution. We also establish sufficient conditions for the nonexistence of positive supersolutions in relationship with the values of tau := lim sup(vertical bar x vertical bar ->infinity) vertical bar x vertical bar b(x) <= infinity. A key ingredient in the proofs is a generalized Hardy-type inequality associated to the p-Laplace operator.

Keywords

Nonlinear elliptic equation;Liouville theorem;super solution;convection term

Authors

AGHAJANI, A.; RADULESCU, V.

Released

27. 10. 2023

Publisher

Association des Annales de l'Institut Fourier

ISBN

1777-5310

Periodical

ANNALES DE L INSTITUT FOURIER

Year of study

73

Number

6

State

French Republic

Pages from

2543

Pages to

2566

Pages count

24

URL

https://aif.centre-mersenne.org/articles/10.5802/aif.3576/

Full text in the Digital Library

http://hdl.handle.net/11012/245083

BibTex

@article{BUT187372,
  author="Asodallah {Aghajani} and Vicentiu {Radulescu}",
  title="Positive supersolutions of non-autonomous quasilinear elliptic equations with mixed reaction",
  journal="ANNALES DE L INSTITUT FOURIER",
  year="2023",
  volume="73",
  number="6",
  pages="2543--2566",
  doi="10.5802/aif.3576",
  issn="1777-5310",
  url="https://aif.centre-mersenne.org/articles/10.5802/aif.3576/"
}