Positive ground state solutions for fractional Laplacian system with one critical exponent and one subcritical exponent

Abstract

In this paper, we consider the following fractional Laplacian system with one critical exponent and one subcritical exponent (((− − ∆) ∆)ss u v+ + νv µu = = | | v u | | 2p∗−−12 u v + + λv, λu, x x ∈ ∈ R RNN where (−∆)s is the fractional Laplacian, 0 < s < 1, N > 2s, λ < µν, 1 < p < 2∗ − 1 and 2∗ = N2−N2s is the Sobolev critical exponent. By using the Nehari manifold, we show that there exists a µ0 ∈ (0, 1), such that when 0 < µ ≤ µ0, the system has a positive ground state solution. When µ > µ0, there exists a λµ,ν ∈ [p(µ − µ0)ν, µν) such that if λ > λµ,ν, the system has a positive ground state solution, if λ < λµ,ν, the system has no ground state solution.