On the singular variational problems (Q812472)

Let \(E={\mathcal D}_a^{1,m}(\mathbb R^N)\) is the space of the completion of \(C_0^\infty (\mathbb R^N)\) with respect to the norm \(\| u\| = (\int | x|^{-am}| \nabla u|^m dx )^{1/m}\). Put \[ \begin{aligned} &S(a,b,\lambda_0 )=\inf \biggl\{ \int (| | x|^{-a}\nabla u|^m +\lambda_0 | x|^{-(a+1)m} | u| ^m )dx \Bigl/ \Bigl(\int | | x| ^{-b}u|^p dx\Bigr)^{m/p};\,u\in E, \, u\not \equiv 0\biggr\},\\ &J(u,v)=| | x| ^{-a}\nabla u|^m +\lambda_1 | x|^{-(a+1)m}| u| ^m +| | x| ^{-a}\nabla v| ^m +\lambda_2 | x| ^{-(a+1)m}| v|^m,\\ &\widetilde S(a,b,\lambda_1 , \lambda_2 \} =\inf \biggl\{ \int J(u,v) \,dx\Bigl/ \Bigl(\int | x|^{-bp} | u|^\alpha | v|^\beta \,dx\Bigr)^{m/p};\;u,v\in E,\;[u,v]\not \equiv [0,0]\biggr\}. \end{aligned} \] Here \(N\geq m+1 >2\), \(0\leq a<(N-m)/m, a\leq b<a+1\) and \(p=\alpha +\beta =Nm/[N-m+m(b-a)]\), \(\alpha \geq 1\), \(\beta \geq 1\). The paper shows the existence of minimizer for \(S(a,b,\lambda_0)\) and \(\tilde S(a,b,\lambda_1 ,\lambda_2 )\).