Positive and nodal solutions for a parametric quasilinear $Q$-sub-Laplacian problem with critical exponential growth on the Heisenberg group

Abstract

In this article, we investigate the following modified quasilinear equation with parameter driven by the $Q$-subLaplacian: \begin{align*} \begin{cases} -\Delta_Q u - \Delta_Q\bigl(|u|^{2\alpha}\bigr)\,|u|^{2\alpha-2} u = \lambda f(\xi,u) & \text{in } \Omega, \\[2mm] u = 0 & \text{on } \partial\Omega, \end{cases} \end{align*} where $\Delta_Q(\cdot):= \mathrm{div}_{\mathbb{H}}\bigl(|\nabla_{\mathbb{H}}(\cdot)|^{Q-2}\nabla_{\mathbb{H}}(\cdot)\bigr)$ denotes the $Q$-subLaplacian on the Heisenberg group $\mathbb{H}^N$, $Q=2N+2$ is the homogeneous dimension, $\Omega \subset \mathbb{H}^N$ is a smooth bounded domain, $\lambda>0$, $\alpha > \frac{1}{2}$, and $f$ has critical or subcritical exponential growth of order $\exp\bigl(\beta|t|^{2\alpha Q/(Q-1)}\bigr)$. We prove three results: the existence of a nontrivial positive weak solution in the critical case for all large $\lambda$, and the existence of a least-energy nodal solution with exactly two nodal domains, under subcritical and critical exponential growth. A change of variables $u=g(v)$ reduces the problem to a quasilinear problem whose energy functional is of class $C^1$; the exponent $2\alpha Q/(Q-1)$ arises from the growth of $g$. We handled the exponential growth using the sharp Moser-Trudinger inequality of Cohn and Lu. The positive solution is obtained by the mountain pass theorem and the nodal solutions by minimization on a nodal Nehari set.