Fractional Hamiltonian type system on R with critical growth nonlinearity

This article investigates the existence and properties of ground state solutions to the followingnonlocal Hamiltonian elliptic system: {(-Delta)(1/2)u+V(0)u=g(v),x is an element of R(-Delta)(1/2)v+V(0)v=f(u),x is an element of R, where(-Delta)12is the square root Laplacian operator,V0>0andf,ghave critical exponential growth inR. Using minimization technique over some generalized Nehari manifold, we showthat the set S of ground state solutions is non empty. Moreover for(u,v)is an element of S,u,vareuniformly bounded inL infinity(R)and uniformly decaying at infinity. We also show that the setSis compact inH12(R)xH12(R)up to translations. Furthermore under locally lipschitzcontinuity offandgwe obtain a suitable Poho & zcaron;aev type identity for any(u,v)is an element of S.Wededuce the existence of semi-classical ground state solutions to the singularly perturbed system{is an element of(-Delta)12 phi+V(x)phi=g(psi),x is an element of R is an element of(-Delta)12 psi+V(x)psi=f(phi),x is an element of R, where is an element of>0andV is an element of C(R)satisfy the assumption(V)given below (see Sect.1). Finally as is an element of -> 0, we prove the existence of minimal energy solutions which concentrate around theclosest minima of the potentialV.