@article{Hu2026, 
author = {Jiaqing Hu and QiangQiang Yang},
title = {Multiplicity of solutions to    (  p  ,  q  )-Kirchhoff equations with weight functions via local Palais-Smale condition},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {5},
pages = {13913-13936},
keywords = {(p, q)-Kirchhoff equation, multiple solutions, variational method and critical point theorem, local Palais-Smale condition},
url = {https://www.sciopen.com/article/10.3934/math.2026572},
doi = {10.3934/math.2026572},
abstract = {In this paper, we study the following    (  p  ,  q  )-Kirchhoff equation with weight functions:         {                                                          ∑                          k              ∈              {              p              ,              q              }                                                          (                              −                                  (                                      1                    +                                                                                            ∫                                                                                                                                                                    R                                                            N                                                                                                                                                                            |                                                ∇                        u                                                                                                            |                                                        k                                                                          d                        x                                                                              )                                                                      Δ                    k                                                  u                +                                  |                                u                                                                            |                                                              k                      −                      2                                                                      u                            )                                =          λ          h          (          x          )                      |                    u                                                    |                                            r                −                2                                              u          +          g          (          x          )                      |                    u                                                    |                                            s                −                2                                              u          ,                                              u          ∈                                    W                              1                ,                p                                              (                                                    R                            N                                )          ∩                                    W                              1                ,                q                                              (                                                    R                            N                                )          ,                        where    1  <  q  ≤  p,    {  r  ,  s  }  ⊂  (  2  p  ,  p  ∗  ), and    p  <  N  <  2  p. By applying a critical point theorem following Perera (J. Anal. Math., 2025), there exists              Λ      m        ≥  0 such that the equation has at least    m  ∈      N   pairs of nontrivial solutions for every    λ  >            Λ      m      . Particularly, we need only prove that the corresponding energy functional satisfies the local Palais-Smale condition, and an explicit expression for              Λ      m       is given, which generalizes some results in the existing literature and provides a new perspective for searching for multiple solutions of Kirchhoff-type equations.}
}