Multiplicity and concentration of positive solutions for the fractional Schrödinger–Poisson systems with critical growth

Abstract

In this paper, we study the multiplicity and concentration of solutions for the following critical fractional Schrodinger–Poisson system: \begin{eqnarray*} \left\{ \begin{array}{ll} \epsilon^{2s}(-\triangle)^{s} {u}+ V(x)u+\phi u =f(u)+|u|^{2^*_{s}-2}u &\mbox{in}\,\,\R^3, \\[2.5mm] \epsilon^{2t}(-\triangle)^{t}{\phi}=u^2 &\mbox{in}\,\, \R^3, \end{array} \right. \end{eqnarray*} where ϵ > 0 is a small parameter, (− △ ) α denotes the fractional Laplacian of order α = s,t ∈ (0,1), where 2 α ∗ 6/3−2α is the fractional critical exponent in Dimension 3; V ∈ C 1 (ℝ 3 ,ℝ + ) and f is subcritical. We first prove that for ϵ > 0 sufficiently small, the system has a positive ground state solution. With minimax theorems and Ljusternik–Schnirelmann theory, we investigate the relation between the number of positive solutions and the topology of the set where V attains its minimum for small ϵ . Moreover, each positive solution u ϵ converges to the least energy solution of the associated limit problem and concentrates around a global minimum point of V .