Existence Of Solutions For Several Elliptic Problems

In this paper,using variational methods,we study existence of solutions for several elliptic systems.Firstly,we deal with the existence of solutions for a class of Hamiltonian-Choquard type systems where μ1,μ2(0,2),0<α<μ1/2,0<β<μ2/2,Iμ1 and Iμ2 denote the Riesz potential,*indicates the convolution operator,F(s),G(s)are the primitive of f(s),g(s)with f(s),g(s)have exponential growth in R2 and V(x)is a continuous positive potential.Using the Moser functions and linking theorem,we obtain the existence of solutions for a class of Hamiltonian Choquard-type elliptic systems in the plane with exponential growth involving singular weights.Then,we consider the following class of fractional Hamiltonian systems where(-Δ)1/2 is the square root Laplacian operator,μ1,μ2∈(0,1),Iμ1,Iμ2 denote the Riesz potential,*indicates the convolution operator,F(t),G(t)are the primitive of f(t),g(t)with f(t),g(t)have exponential critical growth in R.Using the linking theorem and variational methods,we establish the existence of at least one positive solution to the above problem.Finally,we study the existence of normalized solutions to the following nonlinear Choquard equation with exponential growth where a>0 is prescribed,λ∈R,α∈(0,2),Iα denotes the Riesz potential,*indicates the convolution operator,the function f(t)has exponential growth in R2 and F(t)=∫0tf(τ)dτ.Using the Pohozaev manifold and variational methods,we establish the existence of normalized solutions to the above problem.