Advances in Differential Equations

Nonexistence results for differential inequalities involving $A$-Laplacian

Agnieszka Kałamajska, Katarzyna Pietruska-Pałuba, and Iwona Skrzypczak

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We give a sufficient condition for nonexistence of nontrivial nonnegative solutions to the partial differential inequality involving $A$-Laplacian: $-\Delta_A u\ge \Phi (u)$, where $u$ is defined on ${{\mathbb{R}^{n}}}$. The condition obtained relies on the rate of decay at infinity of certain functions involving $A$ and $\Phi$. The techniques, based on methods due to Mitidieri and Pohozaev, exploit suitable a priori estimates in the framework of Orlicz-Sobolev spaces. The result is illustrated by logarithmic $A$-Laplacians and logarithmic functions $\Phi$.

Article information

Adv. Differential Equations, Volume 17, Number 3/4 (2012), 307-336.

First available in Project Euclid: 17 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26D10: Inequalities involving derivatives and differential and integral operators 35D30: Weak solutions 35J60: Nonlinear elliptic equations 35R45: Partial differential inequalities


Kałamajska, Agnieszka; Pietruska-Pałuba, Katarzyna; Skrzypczak, Iwona. Nonexistence results for differential inequalities involving $A$-Laplacian. Adv. Differential Equations 17 (2012), no. 3/4, 307--336.

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