The Existence And Asymptotic Behavior Of Solutions To A Singular Dirichlet Problem For The Monge-Ampère Equation

For a smooth,bounded and strictly convex domain ? of Rn(n?2).In this paper,we study the existence,the nonexistence,the global estimation,the global behavior,the regularity and the exact asymptotic behavior of which the parameters in the weight approach their critical values of global solutions to the singular Dirichlet problem for the Monge-Ampère equationdet D2u=b(x)g(-u)(1+|?u|2)q/2,u<0.x??,u|(?)?=0.Our approaches are based on the perturbation method and constructing the suitable sup-and sub-solution.We also combine the principle of comparison,the Karamata regular vatiation theory and the relevant knowledge of asymptotic behavior.where D2u(x)=((?)2u(x)/(?)xi(?)xj)is defined as the Hessian of u(x),det denotes its de-terminant,det D2 is called the Monge-Ampère operator,?u(x)denotes the gradient of u(x),b?C?(?)is a positive function in ?,it may be allowed to be zero on the boundary or to be suitably singular,g?C1((0,?),(0,?)),0?q<n+1.when 0<q<n+1,we study the existence and the global estimation of global solutions to the above problem.When q=0,the above problem becomesdet D2u=b(x)g(-u),u<0,x??,u|(?)?=0,we study the existence,the global behavior,the regularity and the exact asymp-totic behavior of which the parameters in the weight approach their critical values of its global solutions.In the first chapter,we introduce the research background and main results.In the second chapter,we introduce the necessary preparatory knowledge and related lemmas to prove it.In the third chapter,using the above two methods and some related knowledge,we prove the existence and the global estimation of global solutions to the above prob-lem when 0<q<n+1.In the last chapter,when q=0,we study the existence,the nonexistence,the global behavior,the regularity and the exact asymptotic behavior of which the pa-rameters in the weight approach their critical values of global solutions to the above problem.