Infinitely many solutions for a class of elliptic boundary value problems with $(p,q)$-Kirchhoff type

TL;DR

The paper addresses the existence of infinitely many weak solutions for a nonlocal elliptic system of $(p,q)$-Kirchhoff type. By formulating a variational energy functional and verifying boundedness, compactness, and geometric conditions under sub-$(p,q)$ growth, the authors apply a critical point theorem due to Ding to obtain infinitely many solutions. Two main results are established: Theorem 1.1 under $(G2)$ and $(G4)$ and Theorem 1.2 under $(G2)'$ and $(G4)$, both relying on the sub-$(p,q)$ growth framework and the Cerami-type $(C)$-condition. This extends multiplicity results for nonlocal Kirchhoff systems using variational methods and a specialized critical point theorem, with potential implications for nonlinear PDEs with nonlocal operators.

Abstract

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.