Existence And Multiplicity Of Solutions For Several Classes Of Nonlocal Elliptic Equations

The study of nonlocal problems is one of the important topics in partial differential equation(PDE),which has been widely used in many fields,such as physics,chemistry,biology,finance and so on.In this paper,we study the existence and multiplicity of solutions for several classes of nonlocal elliptic equations.The main results include the following four aspectsThe eigenvalue theory of linear operator plays a very important role in the study of corresponding nonlinear problem,especially the existence of solution for nonlinear elliptic differential equation strongly depends on the eigenvalue of corresponding linear operator.In order to study the existence of nontrivial solutions for the following fractional Schr?dinger equation with indefinite potential(-Δ)su+V(x)u=f(x,u),x∈RN,we first consider the corresponding eigenvalue problem(-Δ)su+V(x)u=λu,x∈RN,and obtain that the problem exists a sequence of eigenvalues which converges to infinity Based on this result,we prove that the equation admits a nontrivial solution and infinitely many large energy solutions under the assumption that f satisfies superlinear and sub-critical growth.Moreover,when f(x,u)=|u|2s*-2u+βδ(x)|u|q-2u(β>0,1<q<2),we establish an existence criterion of infinitely many small energy solutions for the aforemen-tioned equation,which extends the result of positive potential to the case of indefinite potentialSecondly,we are concerned with the following fractional Schr?dinger equation with magnetic field in the whole spaceε2s(-Δ)A/εsu+V(x)u=f(|u|2)u,x∈RN,where ε is a positive parameter,ε∈(0,1),N≥ 3,V∈C(RN,R)and A∈C0,α(RN,RN)with α∈(0,1],are the electric and magnetic potentials respectively,V may be sign-changing,f∈C1(R,R)is superlinear but without satisfying the Ambrosetti-Rabinowitz condition.We prove the existence of a nonnegative ground state solution for the corre-sponding limit equation,and then we obtain that the above equation admits a ground state solution,by combining the concentration-compactness principle and mountain pass theorem.Thirdly,we investigate the following fractional Schr?dinger-Poisson system with concave-convex nonlinearities#12 where λ is a parameter,s,α∈(0,1)and 2s+2a>3.For the case of g(x,u)=b(x)|u|q-2u with 1<q<2,under some suitable assumptions on V,f and b,we present that the above system has at least two different nontrivial solutions via the Ekeland’s variational principle and mountain pass theorem.In addition,for general nonlinear term g,we derive the existence of infinitely many large energy solutions for the system via fountain theorem,which extends the result of classical Schr?dinger-Poisson system to the nonlocal case.Finally,to consider the existence of solution for the following general fractional Lapla-cian equation(-Δ)spu=λg(x)|u|p-2u+f(x,u),x∈RN,we study the corresponding eigenvalue problem(-Δ)p,su=λg(x)|u|p-2u,2,x∈RN,where 0<s<1<p<∞ and N>sp.In the case of indefinite weight,we construct a new fractional Sobolev space as the workspace and prove that the problem exists a sequence of eigenvalues which converges to infinity and that the first eigenvalue is simple,the corresponding eigenfunction may be positive in RN.Based on these results,under certain assumptions on ω and h.we show the existence of infinitely many solutions for the problem with f(x,u)=ω(x)|u|q-2u-h(x)|u|r-2u(1<q<p<r).It is worth mentioning that here r may be greater than the critical Sobolev exponent ps*=Np/N-ps.Furthermore,we also get that the above problem with critical nonlinearity has a nontrivial solution.