| Abstrakt: |
Motivated by the Hardy-Sobolev inequality with multiple Hardy potentials, we consider the following minimization problem : $$\begin{align*} & \inf \left\{\vphantom{\frac{|u|^{2^*_s}}{|x|^s}} \int_\Omega |\nabla u|^2 \,{\rm d}x - \lambda_1 \int_\Omega \frac{u^2}{|x-P_1|^2}\,{\rm d}x \right. \\ & \quad - \left.\lambda_2 \int_\Omega \frac{u^2}{|x-P_2|^2}\,{\rm d}x\ \middle| \ u \in H^1_0(\Omega), \ \int_\Omega \frac{|u|^{2^*_s}}{|x|^s}\,{\rm d}x=1 \right\} \end{align*}$$ inf { | u | 2 s ∗ | x | s ∫ Ω | ∇ u | 2 d x − λ 1 ∫ Ω u 2 | x − P 1 | 2 d x − λ 2 ∫ Ω u 2 | x − P 2 | 2 d x | u ∈ H 0 1 (Ω) , ∫ Ω | u | 2 s ∗ | x | s d x = 1 } where $ N \geq ~3 $ N ≥ 3 , Ω is a smooth domain, $ \lambda _1, \lambda _2 \in \mathbb {R} $ λ 1 , λ 2 ∈ R , $ 0, P_1, P_2 \in \Omega $ 0 , P 1 , P 2 ∈ Ω , $ s \in (0,2) $ s ∈ (0 , 2) and $ 2^*_s = \frac {2(N-s)}{N-2} $ 2 s ∗ = 2 (N − s) N − 2 . Concerning the coefficients of Hardy potentials, we derive a sharp threshold for the existence and non-existence of a minimizer. In addition, we study the existence and non-existence of a positive solution to the Euler-Lagrangian equations corresponding to the minimization problems. [ABSTRACT FROM AUTHOR] |
