Existence results for strongly nonlinear elliptic equations of infinite order (Q2460707)

The authors investigate a strongly nonlinear elliptic equation \(Au + g(x,u) = f(x)\), \(x \in \Omega,\) with boundary conditions of Dirichlet type. Here \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(A\) is a nonlinear elliptic operator satisfying some growth and coerciveness conditions, and the nonlinear term \(g(x,u)\) satisfies a sign condition on \(u\). Existence results for generalized Soboloev spaces of finite order and for Soboloev spaces of infinite order are obtained separately. The theorems extend and complement results of to Brézis, Browder and Webb. Several examples are mentioned at the end of the paper.