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} -Δ_Q u - Δ_Q\bigl(|u|^{2α}\bigr)\,|u|^{2α-2} u = λf(ξ,u) & \text{in } Ω, \\[2mm] u = 0 & \text{on } \partialΩ, \end{cases} \end{align*} where ${\Delta }_Q(\cdot ):={\mathrm{div}}_{\mathbb{H}}(|{\nabla }_{\mathbb{H}}(\cdot ){|}^{Q-2}{\nabla }_{\mathbb{H}}(\cdot ))$ 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 $\exp (\beta |t{|}^{2\alpha Q/(Q-1)})$. 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.