Iterative and variational methods for the solvability of some semilinear equations in Hilbert spaces (Q1196642)

This article deals with the nonlinear equation \(Lu=N(u)\), where \(L:{\mathcal D}(L)\subseteq H\to H\) is a selfadjoint operator in a real Hilbert space \(H\), and \(N: H\to H\) is a gradient operator such that the condition \(\langle A(u-v),u-v\rangle\leq\langle N(u)-N(v),u-v\rangle \leq \langle B(u-v),u-v\rangle\) holds for some continuous selfadjoint opertors \(A,B: H\to H\) such that \(B-A\) is positive and \([0,1]\cap\text{sp}(B-A)^{- {1\over 2}}(L-A)(B-A)^{-{1\over 2}}=\emptyset\). An existence and uniqueness theorem is stated, and some explanatory examples are presented. As an application the \(2\pi\)-periodic Dirichlet problem \(\square u-V'(t,x,u)=h(t,x)\) is considered on \(]0,\pi[^ n\) with the potential nonlinearity \(V'(t,x,u)\) for which, with suitable \(A(t,x)\) and \(B(t,x)\), \((A(t,x)(u-v),u-v) \leq (V'(t,x,u)-V'(t,x,v),u-v) \leq (B(t,x)(u-v),u-v)\).