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<p>In this paper, we investigate the existence of weak solutions for a class of critical quasilinear problems involving an operator of mixed order obtained by the sum of a classical <i>p</i>-Laplacian and a fractional <i>p</i>-Laplacian, and with logarithmic perturbation term <i>μ</i>∣<i>u</i>∣<sup><i>q</i>−2</sup><i>u</i> log ∣<i>u</i>∣<sup><i>q</i></sup>. For <i>μ</i> ∈ ℝ \{0}, we obtain the existence and multiplicity of nontrivial weak solutions subject to certain conditions on the exponent <i>q</i> and the sign of parameter <i>μ</i>. Due to the sign ∣<i>u</i>∣<sup><i>q</i>−2</sup><i>u</i> log ∣<i>u</i>∣<sup><i>q</i></sup> being uncertain, some more detailed analysis will eventually be needed.</p>

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Mixed local-nonlocal quasilinear problems with critical growth and logarithmic perturbation

  • 期刊论文

  • 发表日期：

    2026年5月9日

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  • Aliang Xia

摘要

In this paper, we investigate the existence of weak solutions for a class of critical quasilinear problems involving an operator of mixed order obtained by the sum of a classical p-Laplacian and a fractional p-Laplacian, and with logarithmic perturbation term μ∣u∣q−2u log ∣u∣q. For μ ∈ ℝ \{0}, we obtain the existence and multiplicity of nontrivial weak solutions subject to certain conditions on the exponent q and the sign of parameter μ. Due to the sign ∣u∣q−2u log ∣u∣q being uncertain, some more detailed analysis will eventually be needed.