Missing terms in generalized Hardy's inequalities and its applications (Q1890278)

Let us consider the Hardy inequality of the type \[ \int_\Omega|\Delta^l u(x)|^2dx\geq H\int_\Omega| u(x)|^2| x|^{-4l}\,dx \] for \(l=1,2\). It is known that for the inequality with the optimal constant \(H\) there exists no extremal function \(u\) in the Sobolev space \(H^{2l}_0(\Omega)\). Therefore it is natural to consider that there exist certain ``missing terms'' in the right-hand side. The aim of the paper is to improve the Hardy inequalities by finding these missing terms. An example of such improved inequality is as follows: Let \(N>4\) and let \(\Omega\) be a bounded domain in \(\mathbb R^N\). Then for any \(u\in H^2_0(\Omega)\) we have \[ \int_\Omega| \Delta u(x)|^2\,dx\geq \] \[ H\int_\Omega| u(x)|^2| x|^{-2}+\lambda_1(\omega_N/|\Omega|)^{2/N}N(N-4)/2\int_\Omega| u(x)|^2| x|^{-2}dx+\lambda_2(\omega_N/|\Omega|)^{4/N}\int_\Omega| u(x)|^2dx, \] where \(\omega_N\) and \(|\Omega|\) are the \(N\)-dimensional Lebesgue measure of the unit ball and of \(\Omega\), respectively, and \(\lambda_1\), \(\lambda_2\) are the first eigenvalues of certain elliptic problems. The results are then used for studying properties of blow-up solutions of some semilinear boundary value problems of elliptic type.