Acta Mathematica Sinica, English Series ( IF 0.7 ) Pub Date : 2021-02-15 , DOI: 10.1007/s10114-021-0125-z Li Wang , Tao Han , Ji Xiu Wang
In this article, we show the existence of infinitely many solutions for the fractional p-Laplacian equations of Schröodinger-Kirchhoff type equation
$$M\left( {\left[ u \right]_{s,p}^p} \right)( - \Delta )_p^su + V(x){\left| u \right|^{p - 2}}u = \lambda ({I_\alpha }*{\left| u \right|^{p_{s,\alpha }^*}}){\left| u \right|^{p_{s,\alpha }^* - 2}}u + \beta k(x){\left| u \right|^{q - 2}}u,x \in {\mathbb{R}^N},$$where \((-\Delta )_p^s\) is the fractional p-Laplacian operator, [u]s,p is the Gagliardo p-seminorm, 0 < s < 1 < q < p < N/s, α ∈ (0,N), M and V are continuous and positive functions, and k(x) is a non-negative function in an appropriate Lebesgue space. Combining the concentration-compactness principle in fractional Sobolev space and Kajikiya’s new version of the symmetric mountain pass lemma, we obtain the existence of infinitely many solutions which tend to zero for suitable positive parameters λ and β.
中文翻译：
涉及分数p -Laplacian的Schrödinger-Choquard-Kirchhoff方程的无穷多个解
在本文中，我们显示了Schröodinger-Kirchhoff型方程的分数p -Laplacian方程的无穷多个解的存在
$$ M \ left（{\ left [u \ right] _ {s，p} ^ p} \ right）（-\ Delta）_p ^ su + V（x）{\ left | u \ right | ^ {p-2}} u = \ lambda（{I_ \ alpha} * {\ left | u \ right | ^ {p_ {s，\ alpha} ^ *}}）{\ left | u \ right | ^ {p_ {s，\ alpha} ^ *-2}} u + \ beta k（x）{\ left | u \ right | ^ {q-2}} u，x \ in {\ mathbb {R} ^ N}，$$其中\（（-\ Delta）_p ^ s \）是分数p-拉普拉斯算子[ u ] s，p是Gagliardo p -seminorm，0 < s <1 < q < p < N / s，α∈ （0，N），M和V是连续和正函数，并且k（x）是在适当的Lebesgue空间中的非负函数。结合分数Sobolev空间中的浓度紧致原理和Kajikiya的对称山口引理的新版本，我们获得了存在许多个解，这些解对于合适的正参数λ和β趋于零。
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