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Nonlinear Liouville theorems for Grushin and Tricomi operators. (English) Zbl 1040.35012
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).

MSC:
35D05 Existence of generalized solutions of PDE (MSC2000)
35R45 Partial differential inequalities and systems of partial differential inequalities
35J70 Degenerate elliptic equations
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