This paper deals with the existence of multiple solutions for the following Kirchhoff type equations involving p-biharmonic operator: $$\begin{aligned}& -M \biggl( \int_{\varOmega} \bigl( \vert \Delta_{p}u \vert ^{2}+ \vert u \vert ^{p} \bigr)\,dx \biggr) \bigl( \Delta _{p}^{2}u- \vert u \vert ^{p-2}u \bigr) =\lambda f(x) \vert u \vert ^{q-2}u+g(x) \vert u \vert ^{m-2}u,\quad x\in\varOmega, \end{aligned}$$ where Ω is a bounded domain in $\mathbb{R}^{N}$ ( $N>1$ ), $\lambda >0$ , $p, q, m>1$ , M is a continuous function, and the weight functions f and g are measurable. We obtain the existence results by combining the variational method with Nehari manifold and fibering maps.

中文翻译：

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涉及p-双调和Kirchhoff型问题的多重结果

本文讨论了以下涉及p-双谐波算子的Kirchhoff型方程的多重解的存在：$$ \ begin {aligned}＆-M \ biggl（\ int _ {\ varOmega} \ bigl（\ vert \ Delta_ {p} u \ vert ^ {2} + \ vert u \ vert ^ {p} \ bigr）\，dx \ biggr）\ bigl（\ Delta _ {p} ^ {2} u- \ vert u \ vert ^ {p- 2} u \ bigr）= \ lambda f（x）\ vert u \ vert ^ {q-2} u + g（x）\ vert u \ vert ^ {m-2} u，\ quad x \ in \ varOmega ，\ end {aligned} $$，其中Ω是$ \ mathbb {R} ^ {N} $（$ N> 1 $），$ \ lambda> 0 $，$ p，q，m> 1 $的有界域，M是一个连续函数，权重函数f和g是可测量的。我们通过将变分方法与Nehari流形和纤维化图相结合来获得存在结果。