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 $$ \mbox {div } \left ( \vert \nabla_ωu\vert^{p-2}\nabla_ωu \right )+v(x) u^{q-1}+μu^{γ-1}=0, \quad z\in Ω, \quad u \Big \vert_{\partial Ω}=0. $$ 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 $|{\nabla }_{\omega }u|={\left(\omega (x{)}^{2/p}|{\nabla }_{x}u{|}^2+|{\nabla }_{y}u{|}^2\right)}^{\frac{1}{2}};$ 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.