Singular double-phase systems with variable growth for the Baouendi-Grushin operator
The authors study a double-phase system of equations involving a singular term in the reaction, in the whole space \(\mathbb{R}^N\). The principal operator is the Baouendi-Grushin operator with variable coefficient, which is defined by \[\Delta_{G(x,y)}u= \sum_{i=1}^{n}\left(|\nabla_x|^{G(x,y)-2}u_{x_i} \right)_{x_i}+|x|^\gamma \sum_{i=1}^{n}\left(|\nabla_y|^{G(x,y)-2}u_{y_i} \right)_{y_i},\] where \(G:\mathbb{R}^N \to (1,\infty)\) is a continuous function. After discussing some properties concerning this operator, the authors prove the existence of a weak solution to the double-phase system. The result is established by monotonicity arguments. The similar operator in the context of a class of double-phase energy functionals arising in the theory of transonic flows, was previously studied in a bounded domain \(\Omega \subset \mathbb{R}^N\) (\(N\geq 2\)) with smooth boundary \(\partial \Omega\), by the authors et al. [Nonlinearity 32, No. 7, 2481--2495 (2019; Zbl 1419.35056)].
scientific article; zbMATH DE number 7356648
scientific article