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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2412.00762 |
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| _version_ | 1866909410706587648 |
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| author | Wang, Cong Su, Jiabao |
| author_facet | Wang, Cong Su, Jiabao |
| contents | In this paper we confirm that $2^*(γ)=\frac{2(N+γ)}{N-2}$ with $γ>0$ is exactly the critical exponent for the embedding from $H_r^1(\mathbb{R}^N)$ into $L^q(\mathbb{R}^N;|x|^γ)$($N\geqslant 3$) (see \cite{2007SWW-1,2007SWW-2}) and name it as the upper Hénon-Sobolev critical exponent. Based on this fact we study the ground state solutions of critical Hénon equations in $\mathbb{R}^N$ via the Nehari manifold methods and the great idea of Brezis-Nirenberg in \cite{1983BN}. We establish the existence of the positive radial ground state solutions for the problem with one single upper Hénon-Sobolev critical exponent. We also deal with the existence of the nonnegative radial ground state solutions for the problems with multiple critical exponents, including Hardy-Sobolev critical exponents or Sobolev critical exponents or the upper Hénon-Sobolev critical exponents. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_00762 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The existence of ground state solutions for critical Hénon equations in $\mathbb{R}^N$ Wang, Cong Su, Jiabao Analysis of PDEs In this paper we confirm that $2^*(γ)=\frac{2(N+γ)}{N-2}$ with $γ>0$ is exactly the critical exponent for the embedding from $H_r^1(\mathbb{R}^N)$ into $L^q(\mathbb{R}^N;|x|^γ)$($N\geqslant 3$) (see \cite{2007SWW-1,2007SWW-2}) and name it as the upper Hénon-Sobolev critical exponent. Based on this fact we study the ground state solutions of critical Hénon equations in $\mathbb{R}^N$ via the Nehari manifold methods and the great idea of Brezis-Nirenberg in \cite{1983BN}. We establish the existence of the positive radial ground state solutions for the problem with one single upper Hénon-Sobolev critical exponent. We also deal with the existence of the nonnegative radial ground state solutions for the problems with multiple critical exponents, including Hardy-Sobolev critical exponents or Sobolev critical exponents or the upper Hénon-Sobolev critical exponents. |
| title | The existence of ground state solutions for critical Hénon equations in $\mathbb{R}^N$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2412.00762 |