Sofonea, Mircea Une méthode variationnelle pour une classe d’équations non linéaires dans les espaces de Hilbert. (A variational method for a class of nonlinear equations in Hilbert spaces). (French) Zbl 0594.47051 Bull. Math. Soc. Sci. Math. Répub. Soc. Roum., Nouv. Sér. 30(78), 47-55 (1986). Multivalued equations of the form \(Au+\partial \beta (u)\ni f\) in which A is a linear, symmetric and positive definite operator in a Hilbert space are considered. The concept of classical solution \(u_ c\) is introduced and a uniqueness result is proved. Since in general the equation \(Au+\partial \beta (u)\ni f\) has no classical solution, the concept of generalized solution in the Sobolev sense \(u_ S\) is introduced, using a variational method. The connection between the solutions \(u_ c\) and \(u_ S\) is given and also the dependence of the generalized solution in the Sobolev sense with respect to the element f is studied. Finally some classical results in the variational method for linear and positive definite operators are generalized. Cited in 1 Document MSC: 47J05 Equations involving nonlinear operators (general) 47B25 Linear symmetric and selfadjoint operators (unbounded) 35A15 Variational methods applied to PDEs Keywords:Friedrichs extension; proximity map; Multivalued equations; linear, symmetric and positive definite operator in a Hilbert; space; generalized solution in the Sobolev sense; variational method; linear, symmetric and positive definite operator in a Hilbert space PDFBibTeX XMLCite \textit{M. Sofonea}, Bull. Math. Soc. Sci. Math. Répub. Soc. Roum., Nouv. Sér. 30(78), 47--55 (1986; Zbl 0594.47051)