On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity
This work focuses on the Kirchhoff-Schrödinger-Poisson-type system with singular term and critical Sobolev nonlinearity as follows: −a+b∫Ω∣∇u∣pdxΔpu+ϕ∣u∣q−2u=λu−γ+∣u∣p∗−2uinΩ,−Δϕ=∣u∣qinΩ,u=ϕ=0on∂Ω,\left\{\begin{array}{ll}-\left(a+b\mathop{\displaystyle \int }\limits_{\Omega }{| \nabla u| }^{p}{\rm{d...
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De Gruyter
2024-12-01
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| Series: | Advances in Nonlinear Analysis |
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| Online Access: | https://doi.org/10.1515/anona-2024-0050 |
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| author | Yang Baoling Zhang Deli Liang Sihua |
| author_facet | Yang Baoling Zhang Deli Liang Sihua |
| author_sort | Yang Baoling |
| collection | DOAJ |
| description | This work focuses on the Kirchhoff-Schrödinger-Poisson-type system with singular term and critical Sobolev nonlinearity as follows: −a+b∫Ω∣∇u∣pdxΔpu+ϕ∣u∣q−2u=λu−γ+∣u∣p∗−2uinΩ,−Δϕ=∣u∣qinΩ,u=ϕ=0on∂Ω,\left\{\begin{array}{ll}-\left(a+b\mathop{\displaystyle \int }\limits_{\Omega }{| \nabla u| }^{p}{\rm{d}}x\right){\Delta }_{p}u+\phi {| u| }^{q-2}u=\lambda {u}^{-\gamma }+{| u| }^{{p}^{\ast }-2}u\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ -\Delta \phi ={| u| }^{q}\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ u=\phi =0\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega ,\end{array}\right. where Ω\Omega is a bounded domain in RN{{\mathbb{R}}}^{N} with Lipschitz boundary ∂Ω\partial \Omega , 0<γ<10\lt \gamma \lt 1,Δpu=div(∣∇u∣p−2∇u){\Delta }_{p}u={\rm{div}}\left({| \nabla u| }^{p-2}\nabla u), 1<p<q<p*21\lt p\lt q\lt \frac{{p}^{* }}{2}, p*=Np⁄N−p{p}^{* }=Np/N-p is the critical Sobolev exponent, and λ>0\lambda \gt 0. With the Nehari manifold approach, the above problem is discovered to have at least one weak solution. Furthermore, the singular term and critical nonlinearity arise concurrently, which is the main innovation and difficulty of this article. To some extent, we generalize the previous results. |
| format | Article |
| id | doaj-art-dfe4ab8745264de6b87bceb7bec5e1bf |
| institution | OA Journals |
| issn | 2191-950X |
| language | English |
| publishDate | 2024-12-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Advances in Nonlinear Analysis |
| spelling | doaj-art-dfe4ab8745264de6b87bceb7bec5e1bf2025-08-20T01:47:47ZengDe GruyterAdvances in Nonlinear Analysis2191-950X2024-12-011313012610.1515/anona-2024-0050On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearityYang Baoling0Zhang Deli1Liang Sihua2College of Mathematics, Changchun Normal University, Changchun, 130032, P.R. ChinaCollege of Mathematics, Changchun Normal University, Changchun, 130032, P.R. ChinaCollege of Mathematics, Changchun Normal University, Changchun, 130032, P.R. ChinaThis work focuses on the Kirchhoff-Schrödinger-Poisson-type system with singular term and critical Sobolev nonlinearity as follows: −a+b∫Ω∣∇u∣pdxΔpu+ϕ∣u∣q−2u=λu−γ+∣u∣p∗−2uinΩ,−Δϕ=∣u∣qinΩ,u=ϕ=0on∂Ω,\left\{\begin{array}{ll}-\left(a+b\mathop{\displaystyle \int }\limits_{\Omega }{| \nabla u| }^{p}{\rm{d}}x\right){\Delta }_{p}u+\phi {| u| }^{q-2}u=\lambda {u}^{-\gamma }+{| u| }^{{p}^{\ast }-2}u\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ -\Delta \phi ={| u| }^{q}\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ u=\phi =0\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega ,\end{array}\right. where Ω\Omega is a bounded domain in RN{{\mathbb{R}}}^{N} with Lipschitz boundary ∂Ω\partial \Omega , 0<γ<10\lt \gamma \lt 1,Δpu=div(∣∇u∣p−2∇u){\Delta }_{p}u={\rm{div}}\left({| \nabla u| }^{p-2}\nabla u), 1<p<q<p*21\lt p\lt q\lt \frac{{p}^{* }}{2}, p*=Np⁄N−p{p}^{* }=Np/N-p is the critical Sobolev exponent, and λ>0\lambda \gt 0. With the Nehari manifold approach, the above problem is discovered to have at least one weak solution. Furthermore, the singular term and critical nonlinearity arise concurrently, which is the main innovation and difficulty of this article. To some extent, we generalize the previous results.https://doi.org/10.1515/anona-2024-0050kirchhoff-schrödinger-poisson systemfibering methodcritical growthsingular problemnehari manifoldvariation methods35j2035r0335j6035j10 |
| spellingShingle | Yang Baoling Zhang Deli Liang Sihua On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity Advances in Nonlinear Analysis kirchhoff-schrödinger-poisson system fibering method critical growth singular problem nehari manifold variation methods 35j20 35r03 35j60 35j10 |
| title | On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity |
| title_full | On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity |
| title_fullStr | On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity |
| title_full_unstemmed | On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity |
| title_short | On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity |
| title_sort | on kirchhoff schrodinger poisson type systems with singular and critical nonlinearity |
| topic | kirchhoff-schrödinger-poisson system fibering method critical growth singular problem nehari manifold variation methods 35j20 35r03 35j60 35j10 |
| url | https://doi.org/10.1515/anona-2024-0050 |
| work_keys_str_mv | AT yangbaoling onkirchhoffschrodingerpoissontypesystemswithsingularandcriticalnonlinearity AT zhangdeli onkirchhoffschrodingerpoissontypesystemswithsingularandcriticalnonlinearity AT liangsihua onkirchhoffschrodingerpoissontypesystemswithsingularandcriticalnonlinearity |