Nonlinear Liouville theorems for Grushin and Tricomi operators. (Q1412361)

The paper concerns with necessary conditions for solvability of the inequality \[ L(x,y,D_x,D_y)u \geq | x|^{-\theta_1}| y|^{-\theta_2} | u|^q, \quad x\in \mathbb R^d,\;y\in \mathbb R^k, \tag{1} \] where \(L\) is a quasi-homogeneous operator, \[ L(f(\lambda^{\delta_1}.,\lambda ^{\delta_2}.))(x,y)=\lambda^h(Lf)(\lambda^{\delta_1}x, \lambda^{\delta_2}y). \] This class of operators includes degenerated elliptic equations in \(\mathbb R^N\). The authors prove that under suitable assumptions on \(\delta_i, \theta_i, h\) there exists a critical exponent \(q_c\) such that for \( 1< q\leq q_c\) there are no nontrivial solutions of (1).