In this paper, we study the following $ (p, q) $-Kirchhoff equation with weight functions:
$ \begin{equation*} \left\{ \begin{aligned} & \sum\limits_{k \in \{ p, q\} } {\left( {- \left( {1 + \displaystyle {\int}_{{\mathbb{R}^N}} {|\nabla u{|^k}dx} } \right){\Delta _k}u + |u{|^{k - 2}}u} \right)} = \lambda h(x)|u{|^{r - 2}}u + g(x)|u{|^{s - 2}}u, \\ & u \in {W^{1, p}}({\mathbb{R}^N}) \cap {W^{1, q}}({\mathbb{R}^N}), \end{aligned} \right. \end{equation*} $
where $ 1 < q \le p $, $ \{ r, s\} \subset (2p, p*) $, and $ p < N < 2p $. By applying a critical point theorem following Perera (J. Anal. Math., 2025), there exists $ {\Lambda _m} \ge 0 $ such that the equation has at least $ m \in \mathbb{N} $ pairs of nontrivial solutions for every $ \lambda > {\Lambda _m} $. Particularly, we need only prove that the corresponding energy functional satisfies the local Palais-Smale condition, and an explicit expression for $ {\Lambda _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.
Citation: Jiaqing Hu, QiangQiang Yang. Multiplicity of solutions to $ (p, q) $-Kirchhoff equations with weight functions via local Palais-Smale condition[J]. AIMS Mathematics, 2026, 11(5): 13913-13936. doi: 10.3934/math.2026572
In this paper, we study the following $ (p, q) $-Kirchhoff equation with weight functions:
$ \begin{equation*} \left\{ \begin{aligned} & \sum\limits_{k \in \{ p, q\} } {\left( {- \left( {1 + \displaystyle {\int}_{{\mathbb{R}^N}} {|\nabla u{|^k}dx} } \right){\Delta _k}u + |u{|^{k - 2}}u} \right)} = \lambda h(x)|u{|^{r - 2}}u + g(x)|u{|^{s - 2}}u, \\ & u \in {W^{1, p}}({\mathbb{R}^N}) \cap {W^{1, q}}({\mathbb{R}^N}), \end{aligned} \right. \end{equation*} $
where $ 1 < q \le p $, $ \{ r, s\} \subset (2p, p*) $, and $ p < N < 2p $. By applying a critical point theorem following Perera (J. Anal. Math., 2025), there exists $ {\Lambda _m} \ge 0 $ such that the equation has at least $ m \in \mathbb{N} $ pairs of nontrivial solutions for every $ \lambda > {\Lambda _m} $. Particularly, we need only prove that the corresponding energy functional satisfies the local Palais-Smale condition, and an explicit expression for $ {\Lambda _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.
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Jiaqing Hu, QiangQiang Yang. Multiplicity of solutions to $ (p, q) $-Kirchhoff equations with weight functions via local Palais-Smale condition[J]. AIMS Mathematics, 2026, 11(5): 13913-13936. doi: 10.3934/math.2026572
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