Ricceri, Biagio A general variational principle and some of its applications. (English) Zbl 0946.49001 J. Comput. Appl. Math. 113, No. 1-2, 401-410 (2000). The main result of the paper is the following variational principle type theorem: Let \(X\) be a topological space, and let \(\Phi ,\Psi :X\rightarrow {R}\) be two sequentially l.s.c. functions. Denote by \(I\) the set of all \( \rho >\inf_{X}\Psi \) such that the set \(\Psi ^{-1}(]-\infty ,\rho [)\) is contained in some sequentially compact subset of \(X.\) Assume that \(I\neq \emptyset .\) For each \(\rho \in I\) denote by \(\mathcal{F}_{\rho }\) the family of all sequentially compact subsets of \(X\) containing \(\Psi ^{-1}(]-\infty ,\rho [),\) and put \(\alpha (\rho)=\sup_{K\in \mathcal{F} _{\rho }}\inf_{K}\Phi .\) Then, for each \(\rho \in I\) and each \(\lambda \) satisfying \(\lambda >\inf_{x\in \Psi ^{-1}(]-\infty ,\rho [)}(\Phi (x)-\alpha (\rho))/(\rho -\Psi (x)),\) the restriction of the function \(\Phi +\lambda \Psi \) to \(\Psi ^{-1}(]-\infty ,\rho [)\) has a global minimum. Among several applications it is provided an existence result for nonlinear elliptic equations. Reviewer: Constantin Zălinescu (Iaşi) Cited in 29 ReviewsCited in 281 Documents MSC: 49J27 Existence theories for problems in abstract spaces 49J45 Methods involving semicontinuity and convergence; relaxation 35J65 Nonlinear boundary value problems for linear elliptic equations 47J30 Variational methods involving nonlinear operators Keywords:sequentially lower semicontinuous functions; variational principle; nonlinear elliptic equations PDFBibTeX XMLCite \textit{B. Ricceri}, J. Comput. Appl. Math. 113, No. 1--2, 401--410 (2000; Zbl 0946.49001) Full Text: DOI arXiv References: [2] Schechter, M.; Tintarev, K., Eigenvalues for semilinear boundary value problems, Arch. Rational Mech. Anal., 113, 197-208 (1991) · Zbl 0719.47048 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.