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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