On singular boundary value problems for the Monge-Ampère operator
Let \(\Omega\) be a smooth, strictly convex, bounded domain in \(\mathbb{R}^N\), \(N\geq 2\) and let \(M\) be the Monge-Ampère operator \(Mu= \text{det}(u_{x_i x_j})\). Let \(d_x= \text{dist}(x, \partial \Omega)\). The authors prove that if \(p\) is a smooth positive function in \(\overline \Omega\), then there exists a unique \(C^\infty(\Omega)\) solution \(u\) of \(Mu= p(x) e^u\) such that \(u\to \infty\) as \(x\to \partial\Omega\) and that \(u(x)- \ln d^{- (N+ 1)}_x\) is bounded in \(\Omega\). They also show that there exists a unique \(C^\infty(\Omega)\) solution \(u\) of \(Mu= p(x) u^\gamma\), \(\gamma> N\), such that \(u\to \infty\) as \(x\to \partial \Omega\), and that there are positive constants \(k_1\) and \(k_2\) such that \(k_1 d^{- \alpha}_x\leq u(x)\leq k_2 d^{- \alpha}_x\), for \(x\in \Omega\) and \(\alpha= (N+ 1)/(\gamma- N)\). The results are established by means of a comparison result for the operator \(M\), a subsolution-supersolution approach, and the existence and regularity theory for the operator \(M\). In addition, the authors deduce the existence of a unique \(C^\infty(\Omega)\cap C(\overline\Omega)\) solution to the problem: \(Mu= p(x) (- u)^{- \gamma}\), \(u< 0\), for \(x\in \Omega\) and \(u= 0\) on \(\partial\Omega\), where \(\gamma> 1\), and the existence of negative constants \(c_1\) and \(c_2\) such that \(c_1 d^\alpha_x\leq u(x)\leq c_2 d^\alpha_x\), \(x\in \Omega\), for \(\alpha= (N+ 1)/(N+ \gamma)\).
scientific article; zbMATH DE number 902444