Nonlocal scalar field equations: Qualitative properties, asymptotic profiles and local uniqueness of solutions

Abstract

We study the nonlocal scalar field equation with a vanishing parameter:(Pϵ){(−Δ)su+ϵu=|u|p−2u−|u|q−2uinRNu∈Hs(RN), where s∈(0,1), N>2s, q>p>2 are fixed parameters and ϵ>0 is a vanishing parameter. For ϵ small, we prove the existence and qualitative properties of positive solutions. Next, we study the asymptotic behavior of ground state solutions when p is subcritical, supercritical or critical Sobolev exponent 2⁎=2NN−2s. For p<2⁎, the ground state solution asymptotically coincides with unique positive ground state solution of (−Δ)su+u=up, whereas for p=2⁎ the asymptotic behavior of the solutions is given by the unique positive solution of the nonlocal critical Emden–Fowler type equation. For p>2⁎, the solution asymptotically coincides with a ground-state solution of (−Δ)su=up−uq. Furthermore, using these asymptotic profile of positive solutions, we establish the local uniqueness of positive solution.