In this paper, we employed the Bernstein method to prove some Liouville-type theorems of the equation $ \Delta_{p_1, \cdots, p_r}v+f(v) = 0 $. Here, $ \Delta_{p_1, \cdots, p_r}v: = \mathrm{div}(\sum_{i = 1}^r|\nabla v|^{p_i-2}\nabla v) $. This could be regarded as a natural generalization of the $ p $-Laplacian and the $ (p, q) $-Laplacian. As applications, we derived Liouville-type theorems of positive solutions to some generalized static Fisher-KPP equation, Allen-Cahn equation, static Newell-Whitehead equation, and Lichnerowicz equation.
Citation: Fanqi Zeng, Jin Ban, Peilong Dong, Xiaoqin Ma. Liouville-type theorems for positive solutions to $ \Delta_{p_1, \cdots, p_r}v+f(v) = 0 $ in $ \mathbb{R}^{m} $[J]. Electronic Research Archive, 2025, 33(10): 5990-6011. doi: 10.3934/era.2025266
In this paper, we employed the Bernstein method to prove some Liouville-type theorems of the equation $ \Delta_{p_1, \cdots, p_r}v+f(v) = 0 $. Here, $ \Delta_{p_1, \cdots, p_r}v: = \mathrm{div}(\sum_{i = 1}^r|\nabla v|^{p_i-2}\nabla v) $. This could be regarded as a natural generalization of the $ p $-Laplacian and the $ (p, q) $-Laplacian. As applications, we derived Liouville-type theorems of positive solutions to some generalized static Fisher-KPP equation, Allen-Cahn equation, static Newell-Whitehead equation, and Lichnerowicz equation.
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Fanqi Zeng, Jin Ban, Peilong Dong, Xiaoqin Ma. Liouville-type theorems for positive solutions to $ \Delta_{p_1, \cdots, p_r}v+f(v) = 0 $ in $ \mathbb{R}^{m} $[J]. Electronic Research Archive, 2025, 33(10): 5990-6011. doi: 10.3934/era.2025266
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