Blow-up solutions concentrated along minimal submanifolds for some supercritical elliptic problems on Riemannian manifolds

摘要

Let (M, g) and ({(K, kappa)}) be two Riemannian manifolds of dimensions m and k, respectively. Let ({omega in C^{2} (N), omega > 0}). The warped product ({M times_omega K}) is the (m +  k)-dimensional product manifold ({M times K}) furnished with metric ({g + omega^{2} kappa}) . We prove that the supercritical problem $$- Delta_{g + omega^{2} kappa} u + hu = u^{frac{m+2}{m-2} pm varepsilon} ,quad u > 0,quad {rm in},, (M times_{omega} K, g + omega^{2} kappa)$$has a solution concentrated along a k-dimensional minimal submanifold ({Gamma}) of ({M times_{omega } N}) as the real parameter ({varepsilon}) goes to zero, provided the function h and the sectional curvatures along ({Gamma}) satisfy a suitable condition. Mathematics Subject Classification 35B10 35B33 35J08 58J05 Keywords Supercritical problem concentration along minimal submanifold To Yvonne Choquet-Bruhat