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A general variational principle and some of its applications. (English) Zbl 0946.49001
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.

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
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[1] P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. in Math., 65, Amer. Math. Soc., Providence, RI, 1986.
[2] Schechter, M.; Tintarev, K., Eigenvalues for semilinear boundary value problems, Arch. rational mech. anal., 113, 197-208, (1991) · Zbl 0719.47048
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