A class of solutions for the Neumann problem \(-\Delta u+ \lambda u= u^{(N+2)/ (N-2)}\) (Q1902027)

For a bounded smooth domain \(\Omega \) in \(\mathbb R^{N}\) (\(N\geq 3\)) the author considers the problem (*): \(-\Delta u +\lambda u =u^{(N+2)/(N-2)}\) in \(\Omega\), \(u>0\) in \(\Omega\), and \(\partial u/\partial \nu =0\) on \(\partial \Omega\), where \(\nu \) is the outer normal to \(\partial \Omega\). He proves the existence of a solution to (*) with higher energy. He also studies the shape of the solutions. Results for lower energy solutions have been given by Adimurthi et al.