We study different maximum principles for non-local non-linear operators with non-standard growth that arise naturally in the context of fractional Orlicz–Sobolev spaces and whose most notable representative is the fractional $g$-Laplacian: ${(-{\Delta }_g)}^{s}u\left(x\right)≔\text{p.v}{\int }_{{\mathbb{R}}^n}g\left(\frac{u\left(x\right)-u\left(y\right)}{{|x-y|}^s}\right)\frac{dy}{{|x-y|}^{n+s}},$being $g$ the derivative of a Young function.

We further derive qualitative properties of solutions such as a Liouville type theorem and symmetry results and present several possible extensions and some interesting open questions. These are the first results of this type proved in this setting.

中文翻译：

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分数阶的极大值原理、Liouville 定理和对称性结果 $G$-拉普拉斯算子

我们研究了在分数 Orlicz-Sobolev 空间的背景下自然出现的具有非标准增长的非局部非线性算子的不同最大值原则，其最显着的代表是分数 $G$-拉普拉斯算子： ${(-{\Delta }_G)}^{秒}你\left(X\right)≔\text{光伏}{\int }_{{\mathbb{电阻}}^n}G\left(\frac{你\left(X\right)-你\left(是\right)}{{|X-是|}^{秒}}\right)\frac{d是}{{|X-是|}^{n+秒}},$存在 $G$ Young 函数的导数。

我们进一步推导出解的定性属性，例如 Liouville 型定理和对称性结果，并提出了几种可能的扩展和一些有趣的开放性问题。这些是在此设置中证明的此类结果的第一个结果。