Singular minimization problems
It is considered the existence of ground state solutions of the following two degenerate elliptic equations on \({\mathbb{R}}^N\): \[ -\text{div} (|x|^{-2a} \nabla u)=|x|^{-bp} |u|^{p-2} u, \quad x\in{\mathbb{R}}^N, \] and \[ -\text{div} (|x|^{-2a} \nabla u)+\lambda |x|^{-2(1+a)} u = |x|^{-bp} |u|^{p-2} u,\quad x\in{\mathbb{R}}^N, \] where \(N\geq 3, 0\leq a \leq(N-2)/2, \) \(a\leq b\leq a+1, \) \(p=\frac{2N}{N-2+2(b-a)},\) and \(\lambda\) is a real parameter. On the strength of an inequality established by Caffarelli, Kohn and Nirenberg, the problems under consideration allow variational formulations for the above parameter range, and the corresponding minimization problems with constraints are formulated. It is proved that minimizing sequences for stated minimization problems are relatively compact up to dilation under \(0\leq a<(N-2)/2, a\leq b<a+1\), and some another conditions on \(a,b,\lambda.\)
scientific article; zbMATH DE number 1437329
scientific article