We study the existence and concentration of positive solutions for the following class of fractional p-Kirchhoff type problems:
$$ \left\{\begin{array}{@{}ll} \left(\varepsilon^{sp}a+\varepsilon^{2sp-3}b \,[u]_{s, p}^{p}\right)(-\Delta)_{p}^{s}u+V(x)u^{p-1}=f(u) & \text{in}\ \mathbb{R}^{3},\\ \noalign{ u\in W^{s, p}(\mathbb{R}^{3}), \quad u>0 & \text{in}\ \mathbb{R}^{3}, \end{array}\right.$$
where
ɛ is a small positive parameter,
a and
b are positive constants,
s ∈ (0, 1) and
p ∈ (1, ∞) are such that
$sp \in (\frac {3}{2}, 3)$,
$(-\Delta )^{s}_{p}$ is the fractional
p-Laplacian operator,
f: ℝ → ℝ is a superlinear continuous function with subcritical growth and
V: ℝ
3 → ℝ is a continuous potential having a local minimum. We also prove a multiplicity result and relate the number of positive solutions with the topology of the set where the potential
V attains its minimum values. Finally, we obtain an existence result when
f(
u) =
uq−1 + γ
ur−1, where γ > 0 is sufficiently small, and the powers
q and
r satisfy 2
p <
q <
p*
s ⩽
r. The main results are obtained by using some appropriate variational arguments.