Schrödinger-Hardy systems involving two Laplacian operators in the Heisenberg group
The authors study a Schrödinger-Hardy PDE system on the Heisenberg group \(\mathbb{H}^n\) of real dimension \(2n+1\), of the form \[ \begin{cases} - \Delta^m u + a(q) |u|^{m-2} - \mu \psi^m \frac{|u|^{m-2} u}{r(q)^m} = H_u(q, u, v) \\ - \Delta^p v + b(q) |v|^{p-2} - \sigma \psi^p \frac{|v|^{p-2} v}{r(q)^p} = H_v(q, u, v). \end{cases}\eqno{(*)} \] Here: {\parindent=0.7cm \begin{itemize}\item[{\(\bullet\)}] \(u,v : \mathbb{H}^n \to \mathbb{R}\) are real-valued functions of \(q \in \mathbb{H}^n\) \item[{\(\bullet\)}] \(\Delta^{\wp}\) is the horizontal \(\wp\)-Laplacian operator on the Heisenberg group given by \(\Delta^\wp u = \text{div}(|Du|^{\wp -2} Du)\), where \(D\) is the horizontal gradient and \(\text{div}\) the horizontal divergence \item[{\(\bullet\)}] The exponents \(m,p\) satisfy \(1 < p < Q\) and \(1 < m \leq p < m^* = mQ/(Q-m)\) where \(Q = 2n+2\) is the homogeneous dimension of \(\mathbb{H}^n\) \item[{\(\bullet\)}] \(\mu, \sigma\) are real parameters \item[{\(\bullet\)}] \(r(q)\) is the homogeneous Heisenberg norm, and \(\psi = |Dr|\) is the magnitude of its horizontal gradient \item[{\(\bullet\)}] \(a,b : \mathbb{H}^n \to (0,\infty)\) are positive continuous functions, bounded away from zero and satisfying certain growth conditions at infinity \item[{\(\bullet\)}] \(H : \mathbb{H}^n \times \mathbb{R}^2 \to \mathbb{R}\) satisfies certain conditions on regularity and growth. \end{itemize}} The main result is that, for \(\mu,\sigma\) in an appropriate range, there exists a nontrivial solution \((u,v)\) to the system \((*)\), with \(u,v\) nonnegative. Also, a generalization of \((*)\) involving additional terms on the right side is studied. The proof goes by constructing an Euler-Lagrange functional \(I_{\mu,\sigma}\) whose minimizer will solve \((*)\). The proof also takes advantage of a ``mountain pass geometry'' property of \(I_{\mu,\sigma}\), and compact embeddings of certain function spaces. This paper builds upon [\textit{A. Fiscella} et al., Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 158, 109--131 (2017; Zbl 1371.35079)], in which a similar method was used to prove existence of solutions to an analogous system in Euclidean space.
scientific article; zbMATH DE number 6893936
scientific article