Concentration phenomena for a fractional Schrödinger-Kirchhoff type equation

In this paper, we deal with the multiplicity and concentration of positive solu- tions for the following fractional Schrödinger-Kirchhoff type equation egin{equation*} M left( rac{1}{arepsilon^{3-2s}} iint_{mathbb{R}^{6}} rac{|u(x) - u(y)|^{2}}{|x-y|^{3+2s}} dxdy + rac{1}{arepsilon^{3}} int_{mathbb{R}^{3}} V(x) u^{2} dx ight) [arepsilon^{2s} (-Delta)^{s} u + V(x) u] = f(u) mbox{ in } mathbb{R}^{3}, end{equation*} where $arepsilon>0$ is a small parameter, $sin (rac{3}{4}, 1)$, $(-Delta)^{s}$ is the fractional Laplacian, M is a Kirchhoff function, V is a continuous positive potential, and f is a superlinear continuous function with subcritical growth. By using penalization techniques and Ljusternik-Schnirelmann theory, we investigate the relation between the number of positive solutions with the topology of the set where the potential attains its minimum.