Existence and nonexistence of positive radial solutions of Neumann problems with critical Sobolev exponents (Q1175235)

The authors study positive radial solutions of the semilinear elliptic Neumann problem in a bounded domain \(\Omega\subset \mathbb{R}^n\) \((n\geq 3)\) \[ - \Delta u= u^p+ \lambda\alpha(x) u\quad\text{in }\Omega,\;\partial u/\partial\nu= 0\quad\text{on } \partial\Omega, \] where \(p= (n+ 2)/(n- 2)\) is the critical Sobolev exponent, \(\lambda> 0\). Since for \(\alpha(x)\geq 0\), this problem admits no solution, the authors study existence and nonexistence in the cases where \(\alpha(x)\leq 0\) or \(\alpha\) changes sign.