An abstract version of concentration compactness on Hilbert spaces applies to to actions of non-compact Lie groups. Using the concentration compactness argument we prove existence of solutions for semilinear problems involving sub-Laplacians on the whole Lie group and on their cer-tain non-compact subsets, including minimizers for Sobolev inequalities. The result is stated for any real connected finite-dimensional Lie group.
2000 Mathematics Subject Classifications: 35H20 - 35J20 - 40A30 - 22E30 - 43A85.
Key words: Sublaplacian - Hörmander condition - mountain pass - subelliptic operators - convergence - compactness - concentration.