Nonexistence of positive radial solutions of a quasilinear Neumann problem with a critical Sobolev exponent
The question of the existence of positive nonconstant solutions of the homogeneous Neumann problem for the equation \(-\triangle u = u^p - \lambda u \) is studied. The authors show that for the critical exponent \(p= (n+2)/(n-2)\) and for the case \(n \geq 7\) there exists a constant \(\lambda_0\) such that for \(0 < \lambda < \lambda_0\) this problem does not admit any nonconstant positive and radial solution. To prove this, the authors establish more general result for the \(m\)-Laplacian, namely: Let \(2 \leq m < n, p = \frac{(m-1)n + m}{n-m}\), \(p-1 < q < p\). Then there exists \(\lambda_0 > 0\) such that the problem: \[ -\text{div}(| \nabla u |^{m-2} \nabla u) = u^p - \lambda u^q \quad\text{in }B(1), \] \(u\) is positive and radial in \(B(1)\) and \(\partial u/\partial \nu= 0\) on \(\partial B(1)\) does not admit any nonconstant solution for \(0 < \lambda < \lambda_0\).
scientific article; zbMATH DE number 1096000