Nonlinear PDE with vector fields (Q1591321)

An equation which is close to the Yamabe-type equation \[ \Delta_g u + \langle b,\nabla u\rangle + au = |u|^{4/{(n-2)}}u \] is studied on a compact Riemannian manifold \((V_n,g)\), \(\dim V_n = n\geq 3\), \(b\) is a gradient field, \(a\) is a positive function or zero. A second studied problem has the form \(\varepsilon\Delta_g u +\ldots\), \(\varepsilon\rightarrow 0\). Nontrivial solvability is proved in the first problem (positive or nodal solution), positive solutions and the concentration phenomenon are studied in the second one.