On minimization problems which approximate Hardy Lp inequality Original Research Article

Let Ω be a smooth bounded domain in View the MathML source with 0∈Ω and let p∈(1,∞)⧹{N}. By a classical inequality of Hardy we have View the MathML source, for all View the MathML source, with View the MathML source being the best constant in this inequality. More generally, for View the MathML source such that η⩾0,η≠0 and η(0)=0 we have, for certain values of λ, that View the MathML source, for all View the MathML source. In particular, it follows that there is no minimizer for this inequality. We consider then a family of approximating problems, namely View the MathML source for ε>0, and study the asymptotic behavior, as ε→0, of the positive minimizers {uε} which are normalized by View the MathML source. We prove the convergence View the MathML source in View the MathML source, where View the MathML source is the unique positive solution (up to a multiplicative factor) of the equation View the MathML source in Ω⧹{0}, with u=0 on View the MathML source.