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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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##### References:
 [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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