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

Abstract

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