The Existence And Concentration Phenomenon Of Semiclassical States To Nonlocal Problems With Competing Potential

The existence of solutions and the analysis of the properties of solutions for nonlocal problems are hot topics in the field of nonlinear analysis in recent years.In this paper,the existence,multiplicity and concentration of the solution of fractional Schr?dinger e-quations and fractional Kirchhoff-type equation with competitive potential are studied by the variational method.In chapter 1,we mainly introduce the background,domestic and foreign research status,preparation knowledge and main work of fractional Schr?dinger equations and fractional Kirchhoff-type equation.In chapter 2,we consider the following nonlinear fractional Schr?dinger equations and where ?>0 is a small parameter,s ?(0,1),p ?(2,2s*),2s*:= 2N/N-2s(N>2s)is the fractional Sobolev critical exponent,(-?)s denotes the fractional Laplacian of order s,V,K and Kj(j = 1,2)are linear and nonlinear bound positive potentials functions,respectively.Under some certain assumptions on V,K and Kj(j = 1,2),satisfy Maximum integer m E N,we apply genus theory and the Concentration Compactness Lemma to prove that the problem(0.0.4)and(0.0.5)possess at least m pairs of semiclas-sical positive ground states for ?>0 sufficiently small.Further,when m>2,there are at least 1 is positive solution,1 is negative solution and 2 sign-changing solutions in these solutions.In chapter 3,we study the following fractional Kirchhoff-type equation with critical growth(0.0.6)x ? R3.where M is a continuous and positive Kirchhoff function,?>0 is a parameter,(-?)s is the fractional Laplace operator with 3/4<s<1,2s*:=6/3-2s is the fractional Sobolev critical exponent for 3-dimension,and V(x),W(x)and K(x)are positive poten-tials.Under some assumptions on potentials,we obtain the existence of a positive ground state solution to(0.0.6)for ?>0 small and ? large.Moreover,we show that these ground state solutions concentrating at a special set characterized by potentials.