数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (5): 2061-2074.doi: 10.1007/s10473-023-0508-6

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TWO DISJOINT AND INFINITE SETS OF SOLUTIONS FOR AN ELLIPTIC EQUATION INVOLVING CRITICAL HARDY-SOBOLEV EXPONENTS*

Khalid BOUABID, Rachid ECHARGHAOUI, Mohssine EL MANSOUR   

  1. Department of Mathematics, Faculty of Sciences, Ibn Tofail University, Kenitra, B. P. 133, Morocco
  • 收稿日期:2022-03-01 发布日期:2023-10-25

TWO DISJOINT AND INFINITE SETS OF SOLUTIONS FOR AN ELLIPTIC EQUATION INVOLVING CRITICAL HARDY-SOBOLEV EXPONENTS*

Khalid BOUABID, Rachid ECHARGHAOUI, Mohssine EL MANSOUR   

  1. Department of Mathematics, Faculty of Sciences, Ibn Tofail University, Kenitra, B. P. 133, Morocco
  • Received:2022-03-01 Published:2023-10-25
  • Contact: †Khalid BOUABID, E-mail: bouabid.khalid@uit.ac.ma
  • About author:Rachid ECHARGHAOUI, E-mail:rachid.echarghaoui@uit.ac.ma; Mohssine EL MANSOUR, E-mail:mohssine.elmansour@uit.ac.ma

摘要: In this paper, by an approximating argument, we obtain two disjoint and infinite sets of solutions for the following elliptic equation with critical Hardy-Sobolev exponents $ \left\{\begin{array}{ll} -\Delta u=\mu \vert u \vert ^{2^{*}-2} u +\frac{ \vert u \vert ^{2^{*}(s)-2}u}{ \vert x \vert ^{s}}+ a(x) \vert u \vert ^{q-2} u & \;{\rm in} \; \Omega, \\ u=0 & \;{\rm on} \; \partial \Omega, \end{array}\right.$ where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{N}$ with $0\in \partial \Omega$ and all the principle curvatures of $ \partial \Omega$ at 0 are negative, $a \in \mathcal{C}^{1}(\bar{\Omega}, \mathbb{R^{\ast}}^{+}), $ $ \mu> 0, $ $0<s<2, $ $1<q<2$ and $N > 2\frac{q+1}{q -1}.$ By $2^{*}:=\frac{2 N}{N-2}$ and $2^{*}(s):=\frac{2 (N-s)}{N-2}$ we denote the critical Sobolev exponent and Hardy-Sobolev exponent, respectively.

关键词: Laplacien, critical Sobolev-Hardy exponent, critical Sobolev exponent, infinitely many solutions, Pohozaev identity

Abstract: In this paper, by an approximating argument, we obtain two disjoint and infinite sets of solutions for the following elliptic equation with critical Hardy-Sobolev exponents $ \left\{\begin{array}{ll} -\Delta u=\mu \vert u \vert ^{2^{*}-2} u +\frac{ \vert u \vert ^{2^{*}(s)-2}u}{ \vert x \vert ^{s}}+ a(x) \vert u \vert ^{q-2} u & \;{\rm in} \; \Omega, \\ u=0 & \;{\rm on} \; \partial \Omega, \end{array}\right.$ where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{N}$ with $0\in \partial \Omega$ and all the principle curvatures of $ \partial \Omega$ at 0 are negative, $a \in \mathcal{C}^{1}(\bar{\Omega}, \mathbb{R^{\ast}}^{+}), $ $ \mu> 0, $ $0<s<2, $ $1<q<2$ and $N > 2\frac{q+1}{q -1}.$ By $2^{*}:=\frac{2 N}{N-2}$ and $2^{*}(s):=\frac{2 (N-s)}{N-2}$ we denote the critical Sobolev exponent and Hardy-Sobolev exponent, respectively.

Key words: Laplacien, critical Sobolev-Hardy exponent, critical Sobolev exponent, infinitely many solutions, Pohozaev identity

中图分类号: 

  • 35J60