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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).

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