Existence results for singular p-biharmonic problem with Hardy potential and critical Hardy-Sobolev exponent
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866916412079996928 |
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| author | Singh, Gurpreet |
| author_facet | Singh, Gurpreet |
| contents | In this article, we consider the singular $p-$biharmonic problem involving Hardy potential and citical Hardy-Sobolev exponent. We study the existence of ground state solutions and least energy sign-changing solutions of the following problem \begin{equation*} Δ_{p}^{2} u -λ_{1} \frac{|u|^{p-2}u}{|x|^{2p}}= \frac{|u|^{p_{*}(α)-2}}{|x|^α}u+λ_{2}\Big(|x|^{-β}*|u|^{q}\Big)|u|^{q-2}u \quad\mbox{ in }\R^{N}, \end{equation*} where $p>2$, $2<q< p_{*}(α)$, $λ_{1}>0$, $λ_{2} \in \R$, $α, β\in (0,N)$, $p_{*}(α)=\frac{p(N-α)}{N-2p}$ and $N\geq 5$. Firstly, we study existence of ground state solutions by using the minimization method on the associated Nehari manifold. Then, we investigate the least energy sign-changing solutions by considering the Nehari nodal set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_18041 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence results for singular p-biharmonic problem with Hardy potential and critical Hardy-Sobolev exponent Singh, Gurpreet Analysis of PDEs 35A15, 35B20, 35Q40, 35Q75 In this article, we consider the singular $p-$biharmonic problem involving Hardy potential and citical Hardy-Sobolev exponent. We study the existence of ground state solutions and least energy sign-changing solutions of the following problem \begin{equation*} Δ_{p}^{2} u -λ_{1} \frac{|u|^{p-2}u}{|x|^{2p}}= \frac{|u|^{p_{*}(α)-2}}{|x|^α}u+λ_{2}\Big(|x|^{-β}*|u|^{q}\Big)|u|^{q-2}u \quad\mbox{ in }\R^{N}, \end{equation*} where $p>2$, $2<q< p_{*}(α)$, $λ_{1}>0$, $λ_{2} \in \R$, $α, β\in (0,N)$, $p_{*}(α)=\frac{p(N-α)}{N-2p}$ and $N\geq 5$. Firstly, we study existence of ground state solutions by using the minimization method on the associated Nehari manifold. Then, we investigate the least energy sign-changing solutions by considering the Nehari nodal set. |
| title | Existence results for singular p-biharmonic problem with Hardy potential and critical Hardy-Sobolev exponent |
| topic | Analysis of PDEs 35A15, 35B20, 35Q40, 35Q75 |
| url | https://arxiv.org/abs/2409.18041 |