On Choquard-Kirchhoff Type Critical Multiphase Problem
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909450428743680 |
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| author | Arora, Anupma Dwivedi, Gaurav |
| author_facet | Arora, Anupma Dwivedi, Gaurav |
| contents | In this paper, we obtain the existence of weak solutions to the Choquard-Kirchhoff type critical multiphase problem: \begin{equation*} \left\{\begin{array}{cc}
&-M(φ_{\h}(\lvert{\nabla u}\rvert))div(\lvert{\nabla u}\rvert^{p(x)-2}\nabla u+a_1(x)\lvert{\nabla u}\rvert^{q(x)-2}\nabla u+a_2(x)\lvert{\nabla u}\rvert^{r(x)-2}\nabla u)
& =λg(x)\lvert{u}\rvert^{γ(x)-2}u+θB(x,u)+κ\left(\int_{\q}\frac{F(y,u(y))}{\lvert{x-y}\rvert^{d(x,y)}}\, dy\right) f(x,u) \ \text{in} \ Ω,
& u=0 \ \text{on} \ {\partial Ω}. \end{array}\right. \end{equation*}
The term $B(x,u)$ on the right-hand side generalizes the critical growth. We obtain existence and multiplicity results by establishing certain embedding results and concentration compactness principle along with the Hardy-Littlewood-Sobolev type inequality for the Musielak Orlicz Sobolev space $ W^{1,\mathcal{T}}(\q)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_03595 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Choquard-Kirchhoff Type Critical Multiphase Problem Arora, Anupma Dwivedi, Gaurav Analysis of PDEs 35J15, 35J20, 35D30, 35J60 In this paper, we obtain the existence of weak solutions to the Choquard-Kirchhoff type critical multiphase problem: \begin{equation*} \left\{\begin{array}{cc} &-M(φ_{\h}(\lvert{\nabla u}\rvert))div(\lvert{\nabla u}\rvert^{p(x)-2}\nabla u+a_1(x)\lvert{\nabla u}\rvert^{q(x)-2}\nabla u+a_2(x)\lvert{\nabla u}\rvert^{r(x)-2}\nabla u) & =λg(x)\lvert{u}\rvert^{γ(x)-2}u+θB(x,u)+κ\left(\int_{\q}\frac{F(y,u(y))}{\lvert{x-y}\rvert^{d(x,y)}}\, dy\right) f(x,u) \ \text{in} \ Ω, & u=0 \ \text{on} \ {\partial Ω}. \end{array}\right. \end{equation*} The term $B(x,u)$ on the right-hand side generalizes the critical growth. We obtain existence and multiplicity results by establishing certain embedding results and concentration compactness principle along with the Hardy-Littlewood-Sobolev type inequality for the Musielak Orlicz Sobolev space $ W^{1,\mathcal{T}}(\q)$. |
| title | On Choquard-Kirchhoff Type Critical Multiphase Problem |
| topic | Analysis of PDEs 35J15, 35J20, 35D30, 35J60 |
| url | https://arxiv.org/abs/2501.03595 |