In this paper we study a class of quasilinear elliptic equations with double phase energy and reaction term depending on the gradient. The main feature is that the associated functional is driven by the Baouendi–Grushin operator with variable coefficient. This partial differential equation is of mixed type and possesses both elliptic and hyperbolic regions. We first establish some new qualitative properties of a differential operator introduced recently by Bahrouni et al. (Nonlinearity 32(7):2481–2495, 2019). Next, under quite general assumptions on the convection term, we prove the existence of stationary waves by applying the theory of pseudomonotone operators. The analysis carried out in this paper is motivated by patterns arising in the theory of transonic flows.

在本文中，我们研究了一类具有双相能量和取决于梯度的反应项的拟线性椭圆型方程。主要功能是相关的功能由具有可变系数的Baouendi–Grushin算子驱动。该偏微分方程是混合型的，具有椭圆和双曲区域。我们首先建立了Bahrouni等人最近引入的微分算子的一些新的定性性质。（非线性32（7）：2481-2495，2019年）。接下来，在对流项的相当一般的假设下，我们通过应用伪单调算子理论证明了平稳波的存在。本文进行的分析是由跨音速流动理论中出现的模式引起的。