Multiplicity and concentration of positive solutions for fractional nonlinear Schrodinger equations with critical growth

Abstract

In this article we consider the multiplicity and concentration behavior of positive solutions for the fractional nonlinear Schrödinger equation ε2s (-Δ)su + V(x)u = u2*s-1 + ƒ(u), x ∈ ℝN, u ∈ Hs(ℝN), u(x) > 0, where ε is a positive parameter, s ∈ (0, 1), N > 2s and 2*s = 2N/N-2s is the fractional critical exponent, and ƒ is a C1 function satisfying suitable assumptions. We assume that the potential V(x) ∈ C(ℝN) satisfies infℝN V(x) > 0, and that there exits k points xj ∈ ℝN such that for each j = 1,..., k, V(xj) are strictly global minimum. By using the variational method, we show that there are at least k positive solutions for a small ε > 0. Moreover, we establish the concentration property of solutions as ε tends to zero.