Concentration of positive solutions for a class of fractionalp-Kirchhoff type equations

Abstract

AbstractWe study the existence and concentration of positive solutions for the following class of fractionalp-Kirchhoff type problems:$$ \left\{\begin{array}{@{}ll} \left(\varepsilon^{sp}a+\varepsilon^{2sp-3}b \,[u]_{s, p}^{p}\right)(-\Delta)_{p}^{s}u+V(x)u^{p-1}=f(u) & \text{in}\ \mathbb{R}^{3},\\ \noalign{ u\in W^{s, p}(\mathbb{R}^{3}), \quad u>0 & \text{in}\ \mathbb{R}^{3}, \end{array}\right.$$whereɛis a small positive parameter,aandbare positive constants,s∈ (0, 1) andp∈ (1, ∞) are such that$sp \in (\frac {3}{2}, 3)$,$(-\Delta )^{s}_{p}$is the fractionalp-Laplacian operator,f: ℝ → ℝ is a superlinear continuous function with subcritical growth andV: ℝ3→ 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 potentialVattains its minimum values. Finally, we obtain an existence result whenf(u) =uq−1+ γur−1, where γ > 0 is sufficiently small, and the powersqandrsatisfy 2p<q<p*s⩽r. The main results are obtained by using some appropriate variational arguments.