Degenerate coercive quasilinear elliptic equations with subcritical or critical exponents in $ \mathbb{R}^N $

Yaotian Shen; Youjun Wang

doi:10.3934/cpaa.2020197

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2020, Volume 19, Issue 9: 4667-4697. Doi: 10.3934/cpaa.2020197

This issue Previous Article Existence and nonexistence of positive radial solutions for a class of $p$-Laplacian superlinear problems with nonlinear boundary conditions Next Article Hardy inequalities for the fractional powers of the Grushin operator

Degenerate coercive quasilinear elliptic equations with subcritical or critical exponents in $ \mathbb{R}^N $

Yaotian Shen and
Youjun Wang^,

School of Mathematics, South China University of Technology, , Guangzhou 510640, China

^*Corresponding author
^*Corresponding author

Received: August 31, 2019

Revised: February 29, 2020

Published: June 2020

Supported by the Fundamental Research Funds for the Central Universities (No.2018MS59) and Natural Science Foundation of Guangdong (No.2018A0303130196)

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Abstract

We study the existence of positive solutions of the following degenerate coercive quasilinear elliptic equations:

$ \!-\text{div}(g^2(u)\nabla u)\!+\!\lambda g(u)g'(u)|\nabla u|^2\!+\!V(x)u\! = \!\beta u^{(1-\gamma)(2^*-1)}\!+\!f(u), \ x\!\in\! \mathbb{R}^N, $

where $ g(t)\in C(\mathbb{R}, \mathbb{R}) $, $ V(x)\in C(\mathbb{R}^N, \mathbb{R}) $, $ \lambda, \gamma \in \mathbb{R} $, $ \beta\geq 0 $ and $ 2^* = \frac{2N}{N-2} $, $ N\geq3 $. The novelty of this paper is that $ g(t) $ is non-increasing with respect to $ |t| $ and $ \lim_{|t|\rightarrow +\infty} g(t) = 0. $ The main results of this paper can be regarded as a supplement to the case that $ g(t) $ is non-decreasing with respect to $ |t| $ which has been extensively studied recently.

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Mathematics Subject Classification: Primary: 35J20, 35J62; Secondary: 35Q55.

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