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I would like to ask you what is the rigorous definition of nonlinear elliptic operator. I checked several books, such as the book by Serrin and Pucci (The Maximum Principle) but it is too advanced for me.

In particular, how do you prove that the fractional $p$-Laplacian operator defined by $$ (-\Delta)_p^s \varphi(x) =-\int_{\mathbb R^N} \dfrac{|\varphi(x)-\varphi(y)|^{p-2} [\varphi(x)-\varphi(y)]}{|x-y|^{N+ps}}dy $$ along each function $\varphi \in C_c^\infty(\Omega)$, where $\Omega$ is open in $\mathbb R^N$, is elliptic?

Please, could you illustrate the basic idea of the required computations?

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    Instead of answering the question, I would point out that looking for a _rigorous definition of a nonlinear elliptic operator_ could be pointless. There is very little chance that you encounter a general theorem about nonlinear elliptic operators, so why define it rigorously in the first place? Note that $p$-Laplace operator itself (i.e. for $s=1$) is only degenerate elliptic, not elliptic in the usual sense. – 2017-01-19

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