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A generalization of the saddle point method with applications. (English) Zbl 0780.35001

The saddle theorem considers a continuous \(G\) differentiable functional in a Hilbert space \(H\), satisfying \(\inf\{G(v)\), \(v\in M\}=m_ 0>- \infty\), \(\limsup G(v)=m<+\infty\) for \(v\in N\) and \(\| v\|\to+\infty\) where \(H=M+N\) is an orthogonal decomposition of \(H\) into closed subspaces \(M\) and \(N\) with \(\dim N<+\infty\). The theorem states that if \(m<m_ 0\) there exist \(c\geq m_ 0\) and a sequence \((u_ k)\) in \(H\) such that \(G(u_ k)\to c\) and \(G'(u_ k)\to 0\).
In this paper the author shows that the condition \(m<m_ 0\) is unnecessary and obtains the same result even if \(m\geq m_ 0\). Some interesting applications on the existence of solutions for boundary value problems are also given.

MSC:

35A15 Variational methods applied to PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
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