Study On Fractional Laplacian Equation With The Hénon Term

In this paper,we are concerned with the existence of multiple solutions for subcritical fractional Laplacian equation of Henon type on the annulus,and the existence of solutions for critical fractional Laplacian equation of Henon type on a bounded domain.In chapter 1,the research background and the main results of this paper are introduced.In chapter 2,the existence of multiple solutions for the fractional Laplacian equation of Henon type#12 is studied,where ?={x?RN|1<|x|<3} is annulus,N?3,?>0,0<?<2,2<p<2?*,2?*=2N/N-? is critical Sobolev exponent.We also discuss the symmetry breaking phenomenon of solutions when ? is sufficiently large or p tends to 2?*and it shows that the ground state solutions cannot be radial functions.In chapter 3,we discuss the existence of solutions for critical fractional Lapla-cian equation of Henon type#12 where ?(?)RN is bounded domain,N?4,0<?<2,?>0,0<?<A1,?1 is the first eigenvalue of the eigenvalue problem corresponding to the above equation.We can use the variational method to prove that this equation has at least one solution.