Remarks on comparison principles for p-Laplacian with extension to (p,q)-Laplacian

TL;DR

This paper extends comparison principles from p-Laplacian operators to a broader class including (p,q)-Laplacian, showing that adding a q-Laplacian relaxes conditions on lower order terms.

Contribution

It generalizes recent comparison principles to (p,q)-Laplacian operators, reducing assumptions on Hamiltonians with polynomial growth in the gradient.

Findings

Adding a q-Laplacian relaxes assumptions on lower order terms.

Comparison principles are extended to a wider class of quasilinear equations.

Results apply to Hamiltonians with polynomial growth depending on x and u.

Abstract

Our purpose is to generalize some recent comparison principles for operators driven by p-Laplacian to a wide class of quasilinear equations including (p, q)-Laplacian. It turns out, in particular, that adding a q-Laplacian to p-Laplacian allows to weaken the assumptions needed on the Hamiltonian of lower order terms. The results are specialized in the case that the Hamiltonian has at most polynomial growth in the gradient with coefficients depending on x and u.