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On Critical Schrödinger–Kirchhoff-Type Problems Involving the Fractional p -Laplacian with Potential Vanishing at Infinity
Mediterranean Journal of Mathematics ( IF 1.1 ) Pub Date : 2020-11-15 , DOI: 10.1007/s00009-020-01619-y
Nguyen Van Thin , Mingqi Xiang , Binlin Zhang

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:

$$\begin{aligned}&M\left( \iint _{{\mathbb {R}}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\mathrm{dxdy}+\int _{{\mathbb {R}}^N}V(x)|u(x)|^{p}\mathrm{{d}}x\right) ((-\Delta )_p^{s}u(x)+V(x)|u|^{p-2}u)\\&\quad =K(x)(\lambda f(x,u)+|u|^{p_s^{*}-2}u), \end{aligned}$$

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 \(sp<N,\) \(p_s^{*}=Np/(N-ps),\) KV 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).



中文翻译:

涉及分数p-Laplacian且无穷大可能消失的临界Schrödinger-Kirchhoff型问题

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:

$$ \ begin {aligned}&M \ left(\ iint _ {{\ mathbb {R}} ^ {2N}} \ frac {| u(x)-u(y)| ^ {p}} {| xy | ^ {N + sp}} \ mathrm {dxdy} + \ int _ {{\ mathbb {R}} ^ N} V(x)| u(x)| ^ {p} \ mathrm {{d}} x \右)((-\ Delta _p ^ {s} u(x)+ V(x)| u | ^ {p-2} u)\\&\ quad = K(x)(\ lambda f(x, u)+ | u | ^ {p_s ^ {**-2} u),\ end {aligned} $$

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)。

更新日期:2020-11-15
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