Existence of radial solutions of the Kohn–Laplacian problem

ABSTRACT

The existence of at least one positive radial solution of the generalized Kohn–Laplacian problem (1) $\begin{array}{rl}-{\mathrm{\Delta }}_{{\mathbb{H}}^n}u+R\left(\xi \right)u& =\sum _{i=1}^k{a}_i\left(|\xi {|}_{{\mathbb{H}}^n}\right)|u{|}^{p_i-2}u\\ & -\sum _{j=1}^m{b}_j\left(|\xi {|}_{{\mathbb{H}}^n}\right)|u{|}^{q_j-2}u\xi \in \mathrm{\Omega },\\ u>0& \xi \in \mathrm{\Omega },\\ \frac{\mathrm{\partial }u}{\mathrm{\partial }n}=0& \xi \in \mathrm{\partial }\mathrm{\Omega },\end{array}$(1) is proved, where ${\mathrm{\Delta }}_{{\mathbb{H}}^n}$ is the Kohn–Laplacian (Heisenberg–Laplacian) operator, Ω is a Korányi ball, $a_i,1\le i\le k$ and $b_j,1\le j\le m$ are nonnegative radial functions and $R\left(\xi \right)$ satisfies some suitable conditions.

KEYWORDS:

• Heisenberg group
• variational principle
• supercritical Neumann problem
• mountain pass geometry theorem

AMS SUBJECT CLASSIFICATIONS:

• 35R03
• 35J20
• 46E35

Disclosure statement

No potential conflict of interest was reported by the author(s).