An existence result for nonliner elliptic problems involving critical Sobolev exponent (Q1088056)

The authors study the problem \[ -\Delta u-\lambda u=| u|^{2^*-2}u\quad on\quad \Omega;\quad u=0\quad on\quad \partial \Omega, \] where \(\Omega\) is a bounded domain in \({\mathbb{R}}^ n\), \(\lambda\) is a real parameter, \(2^*=2n/(n-2)\) is the critical Sobolev exponent for the embedding \(H^ 1_ 0(\Omega)\subset L_ p(\Omega)\). It is proved that in the case \(n\geq 4\), for any \(\lambda\geq 0\) there exists at least one nontrivial solution \(u\in H^ 1_ 0(\Omega)\).