Concentration of positive solutions for a class of fractional<i>p</i>-Kirchhoff type equations
Vincenzo Ambrosio, Teresa Isernia, Vicenţiu D. Rădulescu
Abstract
Abstract 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 ) = u q −1 + γ u r −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.