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Liouville theorems for elliptic problems in variable exponent spaces

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  • We investigate nonexistence of nonnegative solutions to a partial differential inequality involving the p(x){Laplacian of the form

    $- {\Delta _{p(x)}}u \geqslant \Phi (x,u(x),\nabla u(x))$

    in ${\mathbb{R}^n}$, as well as in outer domain $\Omega \subseteq {\mathbb{R}^n}$, where Φ(x; u; ∇u) is a locally integrable Carathéodory’s function. We assume that Φ(x; u; ∇u) ≥ 0 or compatible with p and u. Growth conditions on u and p lead to Liouville-type results for u.

    Mathematics Subject Classification: 26D10, 35J60, 35J91.


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