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A positive solution for an anisotropic $ (p,q) $-Laplacian

  • *Corresponding author: Giovany M. Figueiredo

    *Corresponding author: Giovany M. Figueiredo
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  • Here, the anisotropic $ (p, q) $-Laplacian

    $ - \sum\limits_{i = 1}^N\frac{\partial}{\partial x_i}\left( \left|\frac{\partial u}{\partial x_i}\right|^{p_i-2}\frac{\partial u}{\partial x_i}\right) - \sum\limits_{i = 1}^N\frac{\partial}{\partial x_i}\left( \left|\frac{\partial u}{\partial x_i}\right|^{q_i-2}\frac{\partial u}{\partial x_i}\right) = \lambda u^{\gamma-1} $

    is considered, where $ \Omega $ is a bounded and regular domain of $ \mathbb{R}^N $, $ q_i\leq p_i $ for $ i = 1, \cdots, N $ and $ \gamma > 1 $. The existence of positive solution is proved via sub-supersolution method.

    Mathematics Subject Classification: Primary: 35J25, 35B65, 35J70, 46E35; Secondary: 35K65, 35B45, 35K20.


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