Existence for asymptotically coercive nonlinear elliptic equations in Hilbert spaces (Q1322531)

The paper deals with elliptic equations in a Hilbert space \(H\), written in the form \((L - Q(x))x + e = 0\), where \(L\) is a self-adjoint elliptic operator with domain \(H_1 \subseteq H\), \(Q(x)\) is a bounded self- adjoint operator on \(H\) for each \(x \in H\), and \(e \in H\). The operator \(L\) is, in general, an abstract operator on \(H\), and not necessarily the differential operator appearing in classical problems. The operator \(Q(x)\) satisfies \(Q(x) \geq \lambda_nI\) for large \(|x |\), with \(\lambda_n\) an eigenvalue, as well as an asymptotic coerciveness-type condition. Under this kind of conditions, the equation under consideration has solutions in \(H_1\). There are two main existence results for the abstract case, with applications to such nonlinear equations as \(x'' + f(x)x' + g(t,x)x = e(t)\), with periodic boundary value conditions.