On the study of $(p, Q)$-Laplace Choquard equations with critical Trudinger-Moser nonlinearity in $\mathbb{H}^N$

TL;DR

This paper investigates the existence and multiplicity of solutions for a class of $(p, Q)$-Laplace equations with critical exponential nonlinearity in the Heisenberg group, employing advanced variational methods and inequalities.

Contribution

It introduces new analytical techniques, including a novel Brézis-Lieb lemma, to handle critical exponential nonlinearities in the context of $(p, Q)$-Laplace equations on the Heisenberg group.

Findings

Existence of nontrivial solutions established.

Multiple solutions demonstrated under certain conditions.

New inequalities and lemmas developed for this class of problems.

Abstract

This paper deals with the existence and multiplicity of nontrivial solutions for $(p, Q)$-Laplace equations with the Stein-Weiss reaction under critical exponential nonlinearity in the Heisenberg group $\mathbb{H}^N$. In addition, a weight function and two positive parameters have also been included in the nonlinearity. The developed analysis is significantly influenced by these two parameters. Further, the mountain pass theorem, the Ekeland variational principle, the Trudinger-Moser inequality, the doubly weighted Hardy-Littlewood-Sobolev inequality and a completely new Br\'ezis-Lieb type lemma for Choquard nonlinearity play key roles in our proofs.