Existence and non-existence of minimizers for Hardy-Sobolev type inequality with Hardy potentials

Abstract

Motivated by the Hardy-Sobolev inequality with multiple Hardy potentials, we consider the following minimization problem : $\begin{array}{rl}& inf\{\phantom{\frac{|u{|}^{2_s^{\ast }}}{|x{|}^s}}{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }u{|}^2\mathrm{d}x-{\lambda }_1{\int }_{\mathrm{\Omega }}\frac{u^2}{|x-P_1{|}^2}\mathrm{d}x\\ & -{\lambda }_2{\int }_{\mathrm{\Omega }}\frac{u^2}{|x-P_2{|}^2}\mathrm{d}x|u\in H_0^1(\mathrm{\Omega }),{\int }_{\mathrm{\Omega }}\frac{|u{|}^{2_s^{\ast }}}{|x{|}^s}\mathrm{d}x=1\}\end{array}$where $N\ge 3$, Ω is a smooth domain, ${\lambda }_1,{\lambda }_2\in \mathbb{R}$, $0,P_1,P_2\in \mathrm{\Omega }$, $s\in (0,2)$ and $2_s^{\ast }=\frac{2(N-s)}{N-2}$. 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.

Keywords:

• Semilinear elliptic equation
• existence
• non-existence
• minimizers of Hardy-Sobolev type inequality
• Hardy potential

Mathematic Subject classifications:

• 35J20
• 35J61

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author is supported by NSTC of Taiwan, Grant Number NSTC 110-2115-M-003-019-MY3 and NSTC 111-2218-E-008-004-MBK. The second author is supported by Grant-in-Aid for JSPS Research Fellow (JSPS KAKENHI Grant Number JP19K14571) and Osaka Central Advanced Mathematical Institute: MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849. The third author is supported by the 2020 Yeungnam University Research Grant. The authors thank Professors Futoshi Takahashi and Megumi Sano for their helpful comments on the results.