Nontrivial solutions for Monge-Ampère type operators in convex domains
The authors study the Dirichlet problem \[ \text{det} D^ 2u=f(x,u)\quad\text{in }\Omega,\;u=0\text{ on } \partial\Omega, \tag{*} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^ n\), and \(f\) is a sufficiently smooth function satisfying \(f(x,0)=0\), \(f(x,z)>0\) for \(z<0\). Under these conditions \((*)\) has the trivial solution \(u\equiv 0\). The main result is the existence of a nontrivial convex solution of \((*)\) if \(f\) satisfies appropriate growth conditions as \(z\to 0\) and \(z\to- \infty\). In addition, the eigenvalue problem where \(f\) is replaced by \(\lambda f\) in \((*)\) is investigated, and a nonexistence result for \((*)\) is proved for certain \(f\). These questions have been investigated by a number of authors in recent years. The most general results to date appear to be those of \textit{X.-J. Wang} [J. Differ. Equations 100, No. 1, 95-118 (1992; Zbl 0770.35019)].
existence of a nontrivial convex solution
scientific article; zbMATH DE number 492775