Liouville type theorems for stable solutions of elliptic system involving the Grushin operator

<p style='text-indent:20px;'>We examine the following degenerate elliptic system:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -\Delta_{s} u \! = \! v^p, \quad -\Delta_{s} v\! = \! u^\theta, \;\; u, v&gt;0 \;\;\mbox{in }\; \mathbb{R}^N = \mathbb{R}^{N_1}\times \mathbb{R}^{N_2}, \quad\mbox{where}\;\; s \geq 0\;\; \mbox{and} \;\;p, \theta \!&gt;\!0. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>We prove that the system has no stable solution provided <inline-formula><tex-math id="M1">\begin{document}$ p, \theta &gt;0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ N_s: = N_1+(1+s)N_2&lt; 2 + \alpha + \beta, $\end{document}</tex-math></inline-formula> where</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \alpha = \frac{2(p+1)}{p\theta - 1} \quad\mbox{and} \quad \beta = \frac{2(\theta +1)}{p\theta - 1}. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>This result is an extension of some results in [<xref ref-type="bibr" rid="b15">15</xref>]. In particular, we establish a new integral estimate for <inline-formula><tex-math id="M3">\begin{document}$ u $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ v $\end{document}</tex-math></inline-formula> (see Proposition 1.1), which is crucial to deal with the case <inline-formula><tex-math id="M5">\begin{document}$ 0 &lt; p &lt; 1. $\end{document}</tex-math></inline-formula></p>

Double phase problems with variable growth and convection for the Baouendi–Grushin operator

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.