$p$-biharmonic Kirchhoff equations with critical Choquard nonlinearity
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912562016157696 |
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| author | Goel, Divya Goyal, Sarika Saini, Diksha |
| author_facet | Goel, Divya Goyal, Sarika Saini, Diksha |
| contents | In this article, we deal with the following involving $p$-biharmonic critical Choquard-Kirchhoff equation
$$
\left(a+b\left(\int_{\mathbb R^N}|Δu|^p dx\right)^{θ-1}\right) Δ_{p}^{2}u = α\left(|x|^{-μ}*u^{p^*_μ}\right)|u|^{p^*_μ-2}u+ λf(x) |u|^{r-2} u \; \text{in}\; \mathbb R^N,
$$
where $a\geq 0$, $b> 0$, $0<μ<N$, $N>2p$, $p\geq 2$, $θ\geq1$, $α$ and $λ$ are positive real parameters, $p_μ^{*}= \frac{p(2N-μ)}{2(N-2p)}$ is the upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. The function $f \in L^{t}(\mathbb R^N)$ with $t= \frac{p^{*}}{(p^* -r)}$ if $p<r<p^*:=\frac{Np}{N-2p}$ and $t=\infty$ if $r\geq p^{*}$. We first prove the concentration compactness principle for the $p$-biharmonic Choquard-type equation. Then using the variational method together with the concentration-compactness, we established the existence and multiplicity of solutions to the above problem with respect to parameters $λ$ and \(α\) for different values of $r$. The results obtained here are new even for $p-$Laplacian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_00470 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $p$-biharmonic Kirchhoff equations with critical Choquard nonlinearity Goel, Divya Goyal, Sarika Saini, Diksha Analysis of PDEs 35J20, 35J30, 35J62 In this article, we deal with the following involving $p$-biharmonic critical Choquard-Kirchhoff equation $$ \left(a+b\left(\int_{\mathbb R^N}|Δu|^p dx\right)^{θ-1}\right) Δ_{p}^{2}u = α\left(|x|^{-μ}*u^{p^*_μ}\right)|u|^{p^*_μ-2}u+ λf(x) |u|^{r-2} u \; \text{in}\; \mathbb R^N, $$ where $a\geq 0$, $b> 0$, $0<μ<N$, $N>2p$, $p\geq 2$, $θ\geq1$, $α$ and $λ$ are positive real parameters, $p_μ^{*}= \frac{p(2N-μ)}{2(N-2p)}$ is the upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. The function $f \in L^{t}(\mathbb R^N)$ with $t= \frac{p^{*}}{(p^* -r)}$ if $p<r<p^*:=\frac{Np}{N-2p}$ and $t=\infty$ if $r\geq p^{*}$. We first prove the concentration compactness principle for the $p$-biharmonic Choquard-type equation. Then using the variational method together with the concentration-compactness, we established the existence and multiplicity of solutions to the above problem with respect to parameters $λ$ and \(α\) for different values of $r$. The results obtained here are new even for $p-$Laplacian. |
| title | $p$-biharmonic Kirchhoff equations with critical Choquard nonlinearity |
| topic | Analysis of PDEs 35J20, 35J30, 35J62 |
| url | https://arxiv.org/abs/2509.00470 |