Liouville type theorem for nonlinear elliptic system involving Grushin operator. (English) Zbl 1371.35078
Summary: We study the degenerate elliptic system of the form
\[
\begin{cases} -\Delta_G u = v^p \\ - \Delta_G v = u^q \end{cases}\qquad \text{on}\quad \mathbb{R}^N = \mathbb{R}^{N_1} \times \mathbb{R}^{N_2},
\]
where \(\Delta_G : = \Delta_x + | x |^{2 \alpha} \Delta_y\) is the Grushin operator, \(\alpha \geq 0\) and \(p \geq q > 1\). We establish some Liouville type results for stable solutions of the system. In particular, we prove the comparison principle – a crucial step to establish such results. As consequences, we obtain a Liouville type theorem for the scalar equation and provide a counterpart of the previous result in [C. Cowan, Nonlinearity 26, No. 8, 2357–2371 (2013; Zbl 1277.35159)].
MSC:
| 35J47 | Second-order elliptic systems |
| 35B53 | Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs |
Citations:
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