Infinitely many solutions for a class of elliptic boundary value problems with $(p,q)$-Kirchhoff type
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866913810329108480 |
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| author | Li, Zongxi Qi, Wanting Zhang, Xingyong |
| author_facet | Li, Zongxi Qi, Wanting Zhang, Xingyong |
| contents | In this paper, we investigate the existence of infinitely many solutions for the following elliptic boundary value problem with $(p,q)$-Kirchhoff type
\begin{eqnarray*} \begin{cases}
-\Big[M_1\left(\int_Ω|\nabla u_1|^p dx\right)\Big]^{p-1}Δ_p u_1+\Big[M_3\left(\int_Ωa_1(x)|u_1|^p dx\right)\Big]^{p-1}a_1(x)|u_1|^{p-2}u_1=G_{u_1}(x,u_1,u_2)\ \ \mbox{in }Ω,
-\Big[M_2\left(\int_Ω|\nabla u_2|^q dx\right)\Big]^{q-1}Δ_q u_2+\Big[M_4\left(\int_Ωa_2(x)|u_2|^q dx\right)\Big]^{q-1}a_2(x)|u_2|^{q-2}u_2=G_{u_2}(x,u_1,u_2)\ \ \mbox{in }Ω,
u_1=u_2=0\ \ \quad \quad \quad \quad \quad \quad \quad \ \mbox{ on }\partialΩ.
\end{cases} \end{eqnarray*}
By using a critical point theorem due to Ding in [Y. H. Ding, Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems. Nonlinear Anal, 25(11)(1995)1095-1113], we obtain that system has infinitely many solutions under the sub-$(p,q)$ conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_19576 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Infinitely many solutions for a class of elliptic boundary value problems with $(p,q)$-Kirchhoff type Li, Zongxi Qi, Wanting Zhang, Xingyong Analysis of PDEs In this paper, we investigate the existence of infinitely many solutions for the following elliptic boundary value problem with $(p,q)$-Kirchhoff type \begin{eqnarray*} \begin{cases} -\Big[M_1\left(\int_Ω|\nabla u_1|^p dx\right)\Big]^{p-1}Δ_p u_1+\Big[M_3\left(\int_Ωa_1(x)|u_1|^p dx\right)\Big]^{p-1}a_1(x)|u_1|^{p-2}u_1=G_{u_1}(x,u_1,u_2)\ \ \mbox{in }Ω, -\Big[M_2\left(\int_Ω|\nabla u_2|^q dx\right)\Big]^{q-1}Δ_q u_2+\Big[M_4\left(\int_Ωa_2(x)|u_2|^q dx\right)\Big]^{q-1}a_2(x)|u_2|^{q-2}u_2=G_{u_2}(x,u_1,u_2)\ \ \mbox{in }Ω, u_1=u_2=0\ \ \quad \quad \quad \quad \quad \quad \quad \ \mbox{ on }\partialΩ. \end{cases} \end{eqnarray*} By using a critical point theorem due to Ding in [Y. H. Ding, Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems. Nonlinear Anal, 25(11)(1995)1095-1113], we obtain that system has infinitely many solutions under the sub-$(p,q)$ conditions. |
| title | Infinitely many solutions for a class of elliptic boundary value problems with $(p,q)$-Kirchhoff type |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2504.19576 |