On existence of two positive solutions for the nonlinear subelliptic equations involving nonuniformly p-Laplacian

Abstract

In this paper, we study a solvability result for the nonlinear problem $$ \mbox {div } \left ( \vert \nabla_\omega u\vert^{p-2}\nabla_\omega u \right )+v(x) u^{q-1}+\mu u^{\gamma-1}=0, \quad z\in \Omega, \quad u \Big \vert_{\partial \Omega}=0. $$ assuming for the weight functions $ v \in A_\infty, \, \omega \in A_p $ to belong the Muckenhoupt class and a balance condition of Chanillo-Wheeden's type, with degenerate gradient $\nabla_\omega u =\left ( \omega^{1/p} \nabla_x, \, \nabla_y \right ) $ and its module $ \vert \nabla_\omega u\vert= \left (\omega(x)^{2/p} \vert \nabla_{x}u \vert ^2+\vert \nabla_{y}u\vert^2 \right )^{\frac{1}{2}}; $ the domain $ \Omega\subset \mathbb{R}^N $ is bounded, $ N=n+m, x\in \mathbb{R}^n, \, y\in \mathbb{R}^m$ and $z=(x, y) \in \mathbb{R}^N.$ The range conditions $ q \in (p, pN/(N-p)) $ and $ \gamma \in \left (1, N/(N-1)\right ) $ (or $\gamma\in (1, p)$ and $v^{-\gamma/(q-\gamma)}\in L_{1,loc}(\Omega)$ additionally) and $ \mu \in (0, \Lambda) $ with sufficiently small $ \Lambda $ are assumed also.