Some Compactness Results On Fractional Yamabe Problems

In Euclidean space Rn, the fractional Laplacian operators (-△Rn)γ,γ∈(0,1) were studied extensively in harmonic analysis and stochastic partial differential equa-tions. Since the fractional Laplacian operators (-△Rn)γ are non-local, and the local differential operations can’t work any more. But recently, Caffarelli and Silvestre con-sidered the fractional Laplacian equations to be the equivalent degenerate elliptic e-quations with Dirichlet-to-Neumann boundary condition in one-dimension higher half space R+n+1. Therefore the non-local operator problems were reduced to the locally elliptic differential operator problems. After then, Chang and Gonzalez defined the fractional conformal Laplacian operators by the so-called scattering operators on the asymptotically hyperbolic manifolds, and they also figured out the equivalent degener-ate elliptic equations with Dirichlet-to-Neumann boundary conditions. Using the frac-tional conformal Laplacian operators on asymptotically hyperbolic manifolds, Gonzalez and Qing initiated the research on the fractional Yamabe problems, and for some special cases, they proved the existence of the solutions. In this theses, we also make use of the equivalent degenerate elliptic equations with Dirichlet to Neumann boundary condition. From the view of apriori estimates of the fractional Yamabe problems, we considered some compactness results which are related with the fractional Yamabe problems on the asymptotically hyperbolic Riemannian manifolds. In the Chapter 1, some research background and the main results we obtained were presented. In Chapter 2, the global compactness of the Palais-Smale sequences related with the fractional Yamabe func-tional was showed via the concentration-compactness arguments. The Palais-Smale sequences which, roughly speaking, are the asymptotically weak solutions to the frac-tional Yamabe equations. It had been proved that, after subtracted by a finite number of bubbles, the Palais-Smale sequence converges to the solution of the limit equation in the weighted Sobolev space. Also the energy splitting property of the Palais-Smale sequences and the fact of the non-interference between the bubbles were showed. In Chapter 3, the compactness of the exact solutions to the fractional Yamabe problem-s was proved. On the 4-dimensional asymptotically hyperbolic manifolds, under the assumption that γ∈(3/8,1/2), we proved the compactness of the solutions to the fractional Yamabe problems. We used the contradiction arguments. If the compactness theorem doesn’t hold, by the blow up analysis strategy, the blow up points should be the isolated simple blow up points. And for the isolated simple blow up points, we can use the sign restriction of the Pohozaev identity and the assumption that Positive Mass theorem holds for the fractional conformal Laplacian operators. Then we got the con- tradiction, so the solutions were uniformly bounded, then by the regularity estimates, we had the uniformly lower boundedness. Then the compactness results hold.