We consider the singular Dirichlet problem for the Monge-Ampère type equation{\rm det}\D^2 u=b(x)g(-u)(1+|\nabla u|^2)^{q/2},\u<0,\x\in\Omega,\u|_{\partial\Omega}=0,whereΩis a strictly convex and bounded smoo...We consider the singular Dirichlet problem for the Monge-Ampère type equation{\rm det}\D^2 u=b(x)g(-u)(1+|\nabla u|^2)^{q/2},\u<0,\x\in\Omega,\u|_{\partial\Omega}=0,whereΩis a strictly convex and bounded smooth domain inℝn,q∈[0,n+1),g∈C∞(0,∞)is positive and strictly decreasing in(0,∞)with\lim\limits_{s\rightarrow 0^+}g(s)=\infty,and b∈C∞(Ω)is positive inΩ.We obtain the existence,nonexistence and global asymptotic behavior of the convex solution to such a problem for more general b and g.Our approach is based on the Karamata regular variation theory and the construction of suitable sub-and super-solutions.展开更多
基金supported by Shandong Provincial NSF(ZR2022MA020).
文摘We consider the singular Dirichlet problem for the Monge-Ampère type equation{\rm det}\D^2 u=b(x)g(-u)(1+|\nabla u|^2)^{q/2},\u<0,\x\in\Omega,\u|_{\partial\Omega}=0,whereΩis a strictly convex and bounded smooth domain inℝn,q∈[0,n+1),g∈C∞(0,∞)is positive and strictly decreasing in(0,∞)with\lim\limits_{s\rightarrow 0^+}g(s)=\infty,and b∈C∞(Ω)is positive inΩ.We obtain the existence,nonexistence and global asymptotic behavior of the convex solution to such a problem for more general b and g.Our approach is based on the Karamata regular variation theory and the construction of suitable sub-and super-solutions.