Positive Solutions of Nonuniformly Elliptic Equations with Weighted Convex-Concave Nonlinearity

The paper contains the proof of the existence of two different positive solutions of the problem
$$ \frac{\partial}{\partial z_i}\biggl(a_{ij}(z) \frac{\partial u}{\partial z_j}\biggr)+v(x)u^{q-1}+ \mu u^{p-1}=0, \qquad z\in \Omega, \quad u|_{\partial\Omega}=0, $$
involving convex and concave nonlinearities, the parameter $\mu=\operatorname{const}$, and the variables $z=(x,y) \in \mathbb{R}^n \times \mathbb{R}^{N-n}$. The coefficient matrix $A=\{a_{ij}(z)\}_{i,j=1}^N$ satisfies the nonuniform ellipticity condition
$$ C_1(\omega(x)|\xi|^2+|\eta|^2)\le A(z) \zeta \cdot \zeta \le C_2(\omega(x)|\xi|^2+|\eta|^2) $$
in a bounded domain $\Omega \subset \mathbb{R}^N$, $\zeta=(\xi,\eta) \in \mathbb{R}^n \times \mathbb{R}^{N-n}$, $\zeta \ne 0$. To achieve the goal, the authors consider the conditions on the range of nonlinearity exponents $q \in (2,2N/(N-2))$ and $p\in (1,N/(N-1))$ (or $p\in (1, 2)$ and the additional condition $v^{-p/(q-p)}\in L_1(\Omega)$) and $\mu \in (0,\Lambda)$ for a sufficiently small $\Lambda$; positive weight functions $v\in A_\infty$, $\omega \in A_2$ belong to the corresponding Muckenhoupt classes in the metric of $n$-dimensional Euclidean space and also the balance condition of Chanillo–Wheeden type holds.