Existence And Global Compactness Results Of Solutions To Two Kinds Of Choquard Equations

In this paper7 we mainly study the existence of solutions and global compactness result for two kinds of Choquard equations with upper critical exponent.Firstly,we study the following kind of Choquard equation with upper critical exponent and perturbation term-Δu+u=(Iμ*|u|2μ*)|u|2μ*-2u+λ|u|q-2u x ∈ RN,where Iμ is Riesz potential,μ ∈(0,N),2<q<2*N≥3,2μ*=(2N-μ)/(N-2)is upper critical exponents in the sense of the Hardy-Littlewood-Sobolev inequality.λ is a positive parameter.Firstly,using suitable variational arguments and a nonlocal version of the second concentration-compactness principle,we prove the existence of the mountain-pass solution for λ sufficiently large.Secondly,when λ=1,by using the monotonic trick,global compactness lemma,and variational methods,we establish the existence of ground state solution for above equation.Subsequently,we consider a class of nonlinear fractional p-Laplacian Choquard equation with upper critical exponent(-Δ)psu+a(x)|u|p-2u=(Iμ*|u|pμ,s*)|u|pμ,s*-2u in RN,where 0<μ<N,N≥3.0<s<1<p<N/s,pμ,s*=p(2N-μ)/2(N-sp)is said to be the fractional pLaplacian upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality and(-Δ)ps is the factional p-Laplacian operator.By assuming a(x)>0,a∈LN/sp(RN)we obtain a Struwe type global compactness result for above equation.