Problems at resonance for equations with periodic nonlinearities.
The results in this article concern differential equations with periodic nonlinearities. In the first part, the authors prove an abstract result concerning some equations involving selfadjoint elliptic operators. The method is variational, using a saddle point theorem and properties of perturbations of functionals having saddle points. This result is then applied to the equation \[ -u''-u+g(u)=h(t)+a\sin(t-\phi), \tag{*} \] where \(g\) is a \(C^1\) periodic function with zero mean value and \(h\) is orthogonal to \(\cos t\) and to \(\sin t\). For periodic boundary conditions (for \(u\) and \(u'\)) and \(a\) small enough, the existence of \(2\pi\)-periodic solutions is proved. Also, an application of the abstract result to a Dirichlet boundary value problem on a square is presented. Using the same strategy, the authors obtain multiple solutions for equation \((*)\) in the case \(g(u)=-\sin(u)\).
scientific article; zbMATH DE number 2001960
scientific article