Positive solutions of Ambrosetti-Prodi problems involving the critical Sobolev exponent (Q5937149)

The authors study the Ambrosetti-Prodi problem: NEWLINE\[NEWLINE \Delta u+a u=K u^{N-1} +t r(x) \quad \text{in} M,\qquad u>0\quad \text{in} M,\qquad u=0\quad \text{on} \partial M, NEWLINE\]NEWLINE with a critical Sobolev exponent on a compact Riemannian manifold \((M,g)\) of dimension \(n\geq 3\) (with or without boundary). It is supposed \(r(x)\geq 0,\) \(K>0\) and \(\Delta +a\) to be coercive. They discuss existence and uniqueness of solutions of the above problem for \(r\in C^1\) and \(t>,=\) or \(<\) than some real number \(\lambda^\ast.\)