Concentration-compactness principle for Trudinger–Moser inequalities on Heisenberg groups and existence of ground state solutions

Abstract

Let H n = C n × R be the n-dimensional Heisenberg group, Q = 2n+ 2 be the homogeneous dimension of H n . We extend the well-known concentration-compactness principle on finite domains in the Euclidean spaces of P. L. Lions to the setting of the Heisenberg group H n . Furthermore, we also obtain the corresponding concentrationcompactness principle for the Sobolev space HW 1,Q (H n ) on the entire Heisenberg group H n .Our results improve the sharp Trudinger-Moser inequality on domains of finite measure in H n by Cohn and the second author [8] and the corresponding one on the whole space H n by Lam and the second author [21]. All the proofs of the concentrationcompactness principles in the literature even in the Euclidean spaces use the rearrangement argument and the Polyá-Szegö inequality. Due to the absence of the Polyá-Szegö inequality on the Heisenberg group, we will develop a different argument. Our approach is surprisingly simple and general and can be easily applied to other settings where symmetrization argument does not work. As an application of the concentration-compactness principle, we establish the existence of ground state solutions for a class of Q-Laplacian subelliptic equations on H n :with nonlinear terms f of maximal exponential growth exp αt Q Q−1 as t → +∞. 1 q .

Singular Hardy-Trudinger-Moser inequality and the existence of extremals on the unit disc

“…They confirmed this conjecture indeed holds for any bounded and convex domain in R 2 in [25]. They proved Theorem C via an argument from local inequalities to global ones using the level sets of functions under consideration developed by Lam and Lu in [13,14] (see also [5,16,39]), together with the Riemann mapping theorem.…”

A critical Trudinger-Moser inequality involving a degenerate potential and nonlinear Schrödinger equations

“…the existence and multiplicity results of the equation (1.5) can be found in, e.g., [16,21,33] and the references therein. Their proofs depend crucially on the compact imbeddings given by the coercive potential.…”

“…It is well known that the imbedding H 1 (R 2 ) ↩→ L 2 (R 2 ) is continuous but not compact, even in the class of radial functions. For the existence of nontrivial solutions in this case, one can refer to [4,9,14,21,24,29,34] and the references therein.…”

Critical and Subcritical Anisotropic Trudinger–Moser Inequalities on the Entire Euclidean Spaces

“…e proofs of both critical and subcritical Trudinger-Moser inequalities (3) and ( 4) rely on the Pólya-Szegö inequality and the symmetrization argument. Lam and Lu [9,10] developed a symmetrization-free method to establish the critical Trudinger-Moser inequality (see also Li, Lu, and Zhu [11]) in settings such as the Heisenberg group where the Pólya-Szegö inequality fails. Such an argument also provides an alternative proof of both critical and subcritical Trudinger-Moser inequalities (3) and ( 4) in the Euclidean space.…”