Équations elliptiques non linéaires sur des variétés avec exposant de Sobolev critique. (Nonlinear elliptic equations on manifolds with the critical Sobolev exponent). (French) Zbl 0694.35059
Summary: Let (M,g) be an n-dimensional Riemannian manifold, \(\Delta\) be the Laplace-Betrami operator on M. Let q(x) be a function in \(L^{\infty}(M)\). Consider the problem:
\[
(-\Delta +q)u=u^{(n+2)/(n- 2)},\quad u>0\quad on\quad M.
\]
If \(3\leq n\leq 5\) we establish the existence of u under the necessary and sufficient condition \(\lambda_ 1(-\Delta +q)>0\). The case \(n\geq 6\) is investigated under suitable hypotheses.
MSC:
| 35J60 | Nonlinear elliptic equations |
| 58J99 | Partial differential equations on manifolds; differential operators |
