Profile decomposition and multiple positive solutions for the perturbed CR Yamabe equation on the Heisenberg group

Abstract

In this article, we study an inhomogeneous critical nonlinear equation involving the sub-Laplacian on the Heisenberg group ${\mathbb{H}}^n$. We prove the multiplicity of positive solutions for the critical problem \begin{align*} \mathcal{L}_{\mathbb H^n} u=|u|^{2^\star-2}u+f(ξ) \quad \text{in } \mathbb H^n, \qquad u>0,\quad u\in S^{1,2}(\mathbb H^n), \end{align*} where ${\mathcal{L}}_{{\mathbb{H}}^n}$ is the sub-Laplacian on ${\mathbb{H}}^n$, $2^{\star }=\frac{2Q}{Q-2}$, $Q=2n+2$, $n\ge 1$, $S^{1,2}({\mathbb{H}}^n)$ is the homogeneous Sobolev space on ${\mathbb{H}}^n$, and $f$ is a nontrivial nonnegative functional in the dual space $(S^{1,2}({\mathbb{H}}^n){)}^{\prime }$ satisfying a suitable smallness condition. The above mentioned equation appeared as a perturbation of the CR Yamabe equation on the Heisenberg group. A major difficulty comes from the lack of compactness of the critical Folland-Stein embedding into critical Lebesgue space. To overcome this, we establish a Palais-Smale profile decomposition for the associated energy functional. The obtained Palais-Smale profile decomposition identifies the precise energy levels at which lack of compactness may occur via energy quantization, and shows that every noncompact Palais-Smale sequence decomposes into a finite superposition of weakly interacting bubbles. As a key analytic ingredient, we establish an improved Folland-Stein-Sobolev inequality involving the Morrey norm, which serves as a fundamental interpolation inequality and plays a crucial role in detecting the concentration of noncompact Palais-Smale sequences.

30 pages. Comments are welcome