  In this work, we develop and analyze a Hybrid High-Order (HHO) method for
steady non-linear Leray-Lions problems. The proposed method has several assets,
including the support for arbitrary approximation orders and general polytopal
meshes. This is achieved by combining two key ingredients devised at the local
level: a gradient reconstruction and a high-order stabilization term that
generalizes the one originally introduced in the linear case. The convergence
analysis is carried out using a compactness technique. Extending this technique
to HHO methods has prompted us to develop a set of discrete functional analysis
tools whose interest goes beyond the specific problem and method addressed in
this work: (direct and) reverse Lebesgue and Sobolev embeddings for local
polynomial spaces, $L^{p}$-stability and $W^{s,p}$-approximation properties for
$L^{2}$-projectors on such spaces, and Sobolev embeddings for hybrid polynomial
spaces. Numerical tests are presented to validate the theoretical results for
the original method and variants thereof.
