Existence Of Solutions To Two Kinds Of Fractional Equations

In the first chapter,we briefly introduce the research background and the main results.In the second chapter,we study the following fractional Kirchhoff equation where a,b>0(-△)α is the fractional Laplacian operator with α∈(3/2-3/p,1);Hα(R3)is the fractional Sobolev space;2α*=6/3-2α is the critical Sobolev exponent with 0<μ<2α;g satisfies the Berestycki-Lions-type condition.By using pohozaev identity,truncation technique and the parameter dependent compactness lemma,we prove the existence of positive solutions for small b and large λ.Moreover,we deduce the decay rate of the positive solution as |x|→∞ and its asymptotic behavior as b→0,λ→∞.In the third chapter,we study the following fractional Choquard-Kirchhoff equation where ε>0 is a parameter;s ∈(0,1),a,b>0 are constants,0<μ<2s;(-△)As is the fractional magnetic Laplacian;A:R3→R3 is a magnetic potential;V:R3→R is a positive potential with a local minimum and f is a continuous nonlinearity with subcritical growth.By using the variational methods and the Ljusternik-Schnirelmann theory,we investigate the multiplicity and concentration of positive solutions for the above problem.