The aim of this paper is to study the existence of solutions for critical Schrödinger–Kirchhoff-type problems involving a nonlocal integro-differential operator with potential vanishing at infinity. As a particular case, we consider the following fractional problem:

where \(M:[0, \infty )\rightarrow [0, \infty )\) is a continuous function, \((-\Delta )_p^{s}\) is the fractional p-Laplacian, \(0<s<1<p<\infty \) with \(sp<N,\) \(p_s^{*}=Np/(N-ps),\) K, V are nonnegative continuous functions satisfying some conditions, and f is a continuous function on \({\mathbb {R}}^N\times {\mathbb {R}}\) satisfying the Ambrosetti–Rabinowitz-type condition, \(\lambda >0\) is a real parameter. Using the mountain pass theorem, we obtain the existence of the above problem in suitable space W. For this, we first study the properties of the embedding from W into \(L_{K}^{\alpha }({\mathbb {R}}^N), \alpha \in [p, p_s^{*}].\) Then, we obtain the differentiability of energy functional with some suitable conditions on f. To the best of our knowledge, this is the first existence results for degenerate Kirchhoff-type problems involving the fractional p-Laplacian with potential vanishing at infinity. Finally, we fill some gaps of papers of do Ó et al. (Commun Contemp Math 18: 150063, 2016) and Li et al. (Mediterr J Math 14: 80, 2017).

本文的目的是研究关键Schrödinger–Kirchhoff型问题的解决方案，该问题涉及一个非局部积分微分算子，其势在无穷大处消失。作为特殊情况，我们考虑以下分数问题：

其中\（M：[0，\ infty）\ rightarrow [0，\ infty} \）是连续函数，\（（-\ Delta _p ^ {s} \）是分数p -Laplacian，\（0 <s <1 <p <\ infty \）其中\（sp <N，\） \（p_s ^ {*} = Np /（N-ps），\） K，  V是满足某些条件的非负连续函数，并且f是满足Ambrosetti–Rabinowitz类型条件的\（{\ mathbb {R}} ^ N \ times {\ mathbb {R}} \）上的连续函数，\（\ lambda> 0 \）是实参。利用山口定理，我们在适当的空间W中获得上述问题的存在。为此，我们首先研究从W嵌入到\（L_ {K} ^ {\ alpha}（{\ mathbb {R}} ^ N），\ alpha \ in [p，p_s ^ {*}]中的性质。\）然后，我们在f上具有一些合适的条件下获得了能量泛函的可微性。据我们所知，这是涉及分数p -Laplacian的简并Kirchhoff型问题的第一个存在性结果，并且可能在无穷大时消失。最后，我们填补了doÓ等人论文的空白。（Commun Contemp Math 18：150063，2016）和Li et al。（Mediterr J Math 14:80，2017）。