Existence Of Solutions For Some Nonlocal Elliptic-type Equations

In this thesis,we study the existence,uniqueness,multiplicity and asymptotic properties of solutions for some nonlocal elliptic equations by using the critical point theory and variational analysis of nonlinear analysis.The main results include the following four aspects.Firstly,we are concerned with the following fractional Choquard equation involving magnetic field and critical nonlinearityε2s(-Δ)A/εsu+V(x)u=ε-α(∫RN|u(y)|2s,α*/|x-y|N-α)dy)|u|2s,α*-2u+ε-α(∫RNF(y,|u(y)2)/(|x-y|N-α)dy)f(x,|u|2)u,x∈RN,where ε is a small positive parameter,s ∈(0,1),α∈(0,N),N>max{2μ+4s,2s+(N-α)/2},(-Δ)As is the fractional magnetic Laplacian operator,and 2s,α*=(N+α)/(N-2s)is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality.Under some suitable assumptions on A,V,f,we get the existence of ground state solutions for above problem by applying the Lions’ concentration compactness argument.Secondly,we investigate the following fractional Choquard equation involving potential and L~2-supercritical nonlinearity(-Δ)su+V(x)u+λu=(Iα*F(u))f(u),x ∈RN,on the constraint∫RNu2dx=ρ2,ρ>0,where s ∈(0,1),N>2s,α∈(0,N),(-Δ)s is the fractional Laplacian operator.The parameter λ∈ R arises as Lagrange multiplier with respect to the mass constraint ‖u‖L~2(RN)=ρ.Iα:RN\{0}→R is the Riesz potential defined as Iα{x)=Γ(N-α/2)/Γ(α/2)πN/22α|x|N-α,where Γ is the Gamma function.Under some mild assumptions imposed on V,f,by introducing a stretched functional and using a min-max argument,we construct a bounded Palais-Smale sequence at the linking level and establish the existence of L~2-normalized solution(u,λ)∈ Hs(RN)× R+ for above problem.Thirdly,we study the following Kirchhoff-type equation involving L~2-subcritical or L~2-supercritical nonlinearity-(a+b ∫R3 |▽u|2dx)Δu+λu=|u|p-2u,x ∈ R3,on the constraint fR3 u2dx=ρ2,where ρ,a,b are given positive constants and λ∈R arises as Lagrange multiplier with respect to the mass constraint ‖u‖L~2(R3)2=ρ2.When p ∈(2,10/3)or p∈(14/3,6),using the genus theory,we construct a sequence of min-max energy level for the stretched functional.By using a min-max argument,we obtain a bounded Palais-Smale sequence at each min-max energy level and establish the existence of infinitely many radial L~2-normalized solutions for above problem.Finally,we consider the following fractional Keller-Segel model with Neumann boundary condition and the general nonlinearity where ε is a positive parameter,Ω?RN(N≥ 2)is a smooth bounded domain,and v is the outward unit normal to ?Q.Under the superlinear and subcritical growth assumptions on h,we prove that there exists at least one nonconstant solution uεfor small ε>0 by using mountain pass lemma and the family of solutions{uε}ε>0 is uniformly bounded as ε→ 0 by using a classical Moser iteration.Moreover,under some mild assumptions imposed on h,we build a Pohozaev-type identity to prove the non-existence of weak bounded solution for above problem.Meanwhile,when N≥ 5,we consider the case of Sobolev critical for above problem,that is,h(t)=tN+1/N-1.We prove that there exists a constant ε0>0 such that the above problem admits a minimal energy solution for ε>ε0.Moreover,if Ω is convex,we show that this solution is constant for sufficiently large ε>0.