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<p>We investigate the existence and multiplicity of solutions for a class of Kirchhoff-type equations involving the (<i>p</i>,&#xa0;<i>q</i>)-Laplacian and a critical nonlinear term. To overcome the lack of compactness, we first establish the local Palais–Smale condition via the concentration compactness principle of Lions. By employing the mountain pass theorem combined with some new analytical techniques, we obtain the existence of nontrivial solutions. Furthermore, by applying a critical point theorem proposed by Perera (J Anal Math. 2025. <a href="https://doi.org/10.1007/s11854-025-0389-9">https://doi.org/10.1007/s11854-025-0389-9</a>), we show that the equation admits multiple solutions when <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\lambda \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation> is sufficiently large. Our results extend and improve some earlier results in the literature.</p>

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Existence and multiplicity of solutions to \(\pmb {(p, q)}\)-Kirchhoff-type equations with critical growth in \(\pmb {{\mathbb {R}}^N}\)

  • 期刊论文

  • 发表日期：

    2025年12月26日

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  • Bingbing Qian,
  • Mengqiu Shao

摘要

We investigate the existence and multiplicity of solutions for a class of Kirchhoff-type equations involving the (p, q)-Laplacian and a critical nonlinear term. To overcome the lack of compactness, we first establish the local Palais–Smale condition via the concentration compactness principle of Lions. By employing the mountain pass theorem combined with some new analytical techniques, we obtain the existence of nontrivial solutions. Furthermore, by applying a critical point theorem proposed by Perera (J Anal Math. 2025. https://doi.org/10.1007/s11854-025-0389-9), we show that the equation admits multiple solutions when \(\lambda \) λ is sufficiently large. Our results extend and improve some earlier results in the literature.