The asymptotic behavior of Palais-Smale sequences on manifolds with boundary (Q482357)

Let \((M^n,g)\) be a compact Riemannian manifold with boundary \(\partial M\) and dimension \(n\geq3.\) The author describes the asymptotic behavior of the Palais-Smale sequences associated to the Yamabe-type equations \[ \begin{cases} \Delta_g u=0 & \text{in}\;M,\quad u\in H^1(M),\\ \dfrac{\partial}{\partial \eta_g}u-h_\nu u+ u^{\frac{n}{n-2}}=0 & \text{on}\;\partial M, \end{cases} \] where \(\{h_\nu\}_{\nu\in\mathbb{N}}\) is a sequence of \(C^\infty(\partial M)\) functions, \(\Delta_g\) is the Laplace-Beltrami operator on \(M\) and \(\eta_g\) is the unit inward normal to \(\partial M.\) It is proved that each of those sequences converges to a solution of the limit equation plus a finite number of ``bubbles'' which are obtained by rescaling fundamental solutions of the corresponding Euclidean equations.