Sobolev norm estimates of solutions for the sublinear Emden-Fowler equation (Q2864646)

Let \(\Omega\) be a bounded open connected domain in \({\mathbb R}^N\) (\(N\geq 2\)) and assume that \(p\in (0,1)\). This paper deals with the Emden-Fowler equation \(-\Delta u=|u|^{p-1}u\) in \(\Omega\) subject to the Dirichlet boundary condition \(u=0\) on \(\partial\Omega\). The main result in the present paper investigates the convergence rate of the Sobolev norm of solutions as the volume of the domain converges to zero. The study is done by estimating the first eigenvalue of the Laplace operator in \(H^1_0(\Omega)\) by means of variational arguments.