Critical Sobolev exponent problem in \({\mathbb{R}{}}^ n(n{\geq{}}4)\) with Neumann boundary condition (Q1174586)

The authors consider the problem; \[ -\Delta u=u^{(n+2)/(n- 2)}+\lambda\alpha u \text{ in } \Omega;\;u>0;\quad {\partial u \over \partial\nu}=0 \text{ on } \partial\Omega, \tag{*} \] where \(\alpha\in C^ \infty(\bar\Omega)\) and \(\Omega\) is a bounded domain of \(\mathbb{R}^ n\), \(n\geq4\) with regular boundary. Under some assumptions on the function \(\alpha\) and on the ''flatness'' of the boundary \(\partial\Omega\) the authors prove that problem (*) admits a solution \(u\in C^ 2(\bar\Omega)\) if and only if \(\lambda\) belongs to a suitable interval \((\nu,\lambda(\alpha))\).