Global Bifurcation Properties For A Class Of Hessian Equations

Hessian equations are a class of second order fully nonlinear differential equat-ions, in which Monge-Ampère equations have direct link with Minkowski problemsin differential geometry, and the research about these overwhelmingly enriches thecontents of partial differential equations and differential geometry. On this basis,many similar questions have been proposed and many important results have beengained. A great quantity of scholars has also discussed the Hessian equations, butthe research about a class of Hessian equations in the plane is still a few.In this thesis, we study the eigenvalue and global bifurcation problems for aclass of Hessian equations in the plane. These equations have many good analysisproperties which are not only second order fully nonlinear elliptic differential equat-ions but also concave functions. On this basis, this thesis analyses to get theproperties of the weak solutions and the results about the eigenvalue and globalbifurcation problems for these equations.First, we need to get the existence，uniqueness and local H lder continuity ofthe weak solutions for these equations. For the aim, we first get the comparisonprinciple of the operator by using the comparison principle of second order fullynonlinear elliptic partial differential equations, and then via the maximum principleof the operator Monge-Ampère in the plane and the function approximation of theclassical solutions we get the existence of the weak solutions. Next, we estimate theinterior gradient bound of the classical solutions and by using the maximumprinciple check the H lder continuity of the weak solutions.Second, by using the existence，uniqueness and local H lder continuity of theweak solutions above, we first define a special operator, and then apply the resultsof the theorems after verifying the conditions to get the existence and uniqueness ofthe eigenvalue for these equations, which provide the necessary foundation for thefollowing discuss of the global bifurcation questions.Finally, via the existence and uniqueness of the weak solutions, we define acompletely continuous operator and by using the homology invariance of theLeray-Schauder degree and some global bifurcation theorems get the globalbifurcation properties of these equations with two classes of inhomogeneousperturbations, one of them satisfies a super-linear growth condition near the originand the other one satisfies a sub-linear growth condition near infinity.