On Critical Schrödinger–Kirchhoff-Type Problems Involving the Fractional p-Laplacian with Potential Vanishing at Infinity

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: M∬R2N|u(x)-u(y)|p|x-y|N+spdxdy+∫RNV(x)|u(x)|pdx((-Δ)psu(x)+V(x)|u|p-2u)=K(x)(λf(x,u)+|u|ps∗-2u),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&M\left( \iint _{{\mathbb {R}}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\mathrm{dxdy}+\int _{{\mathbb {R}}^N}V(x)|u(x)|^{p}\mathrm{{d}}x\right) ((-\Delta )_p^{s}u(x)+V(x)|u|^{p-2}u)\\&\quad =K(x)(\lambda f(x,u)+|u|^{p_s^{*}-2}u), \end{aligned}$$\end{document}where M:[0,∞)→[0,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M:[0, \infty )\rightarrow [0, \infty )$$\end{document} is a continuous function, (-Δ)ps\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\Delta )_p^{s}$$\end{document} is the fractional p-Laplacian, 0<s<1<p<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<s<1<p<\infty $$\end{document} with sp<N,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$sp<N,$$\end{document}ps∗=Np/(N-ps),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_s^{*}=Np/(N-ps),$$\end{document}K, V are nonnegative continuous functions satisfying some conditions, and f is a continuous function on RN×R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^N\times {\mathbb {R}}$$\end{document} satisfying the Ambrosetti–Rabinowitz-type condition, λ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda >0$$\end{document} 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 LKα(RN),α∈[p,ps∗].\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{K}^{\alpha }({\mathbb {R}}^N), \alpha \in [p, p_s^{*}].$$\end{document} 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).