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DOI: 10.23952/jnfa.2025.37
Received May 12, 2025; Accepted October 13, 2025; Published December 1, 2025
Abstract. In this paper, we investigate the existence of infinitely many solutions for the following elliptic boundary value problem with (p,q)-Kirchhoff type
![-\Big[M_1\left(\int_\Omega|\nabla u_1|^p dx\right)\Big]^{p-1}\Delta_p u_1+\Big[M_3\left(\int_\Omega a_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 }\Omega,](https://s0.wp.com/latex.php?latex=-%5CBig%5BM_1%5Cleft%28%5Cint_%5COmega%7C%5Cnabla+u_1%7C%5Ep+dx%5Cright%29%5CBig%5D%5E%7Bp-1%7D%5CDelta_p+u_1%2B%5CBig%5BM_3%5Cleft%28%5Cint_%5COmega+a_1%28x%29%7Cu_1%7C%5Ep+dx%5Cright%29%5CBig%5D%5E%7Bp-1%7Da_1%28x%29%7Cu_1%7C%5E%7Bp-2%7Du_1%3DG_%7Bu_1%7D%28x%2Cu_1%2Cu_2%29%5C+%5C+%5Cmbox%7Bin+%7D%5COmega%2C&bg=ffffff&fg=000000&s=0 "-\Big[M_1\left(\int_\Omega|\nabla u_1|^p dx\right)\Big]^{p-1}\Delta_p u_1+\Big[M_3\left(\int_\Omega a_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 }\Omega,")
![-\Big[M_2\left(\int_\Omega|\nabla u_2|^q dx\right)\Big]^{q-1}\Delta_q u_2+\Big[M_4\left(\int_\Omega a_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 }\Omega,](https://s0.wp.com/latex.php?latex=-%5CBig%5BM_2%5Cleft%28%5Cint_%5COmega%7C%5Cnabla+u_2%7C%5Eq+dx%5Cright%29%5CBig%5D%5E%7Bq-1%7D%5CDelta_q+u_2%2B%5CBig%5BM_4%5Cleft%28%5Cint_%5COmega+a_2%28x%29%7Cu_2%7C%5Eq+dx%5Cright%29%5CBig%5D%5E%7Bq-1%7Da_2%28x%29%7Cu_2%7C%5E%7Bq-2%7Du_2%3DG_%7Bu_2%7D%28x%2Cu_1%2Cu_2%29%5C+%5C+%5Cmbox%7Bin+%7D%5COmega%2C&bg=ffffff&fg=000000&s=0 "-\Big[M_2\left(\int_\Omega|\nabla u_2|^q dx\right)\Big]^{q-1}\Delta_q u_2+\Big[M_4\left(\int_\Omega a_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 }\Omega,")
By using a critical point theorem due to Ding in [Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems, Nonlinear Anal. 25 (1995) 1095-1113], we obtain that the system has infinitely many solutions when the nonlinear terms satisfy the asymptotically-(p,q) conditions.
How to Cite this Article:
Z. Li, W. Qi, X. Zhang, Infinitely many solutions for a class of elliptic boundary value problems with (p,q)-Kirchhoff type, J. Nonlinear Funct. Anal. 2025 (2025) 37.