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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
Keywords:
degenerated elliptic equation
Full Text:
References:
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