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Sublevel sets and global minima of coercive functionals and local minima of their perturbations. (English) Zbl 1083.49004
The paper specifies a variational principle of the same author, already published in [B. Ricceri, J. Comput. Appl. Math. 113, No. 1-2, 401–410 (2000; Zbl 0946.49001)].
It is proved that if \(\Phi\) and \(\Psi\) are two sequentially weakly lower semicontinuous functionals on a reflexive real Banach space \(X\), if \(\Psi\) is continuous and coercive, and if for some \(r>\inf_X\Psi\) the weak closure of \(\Psi^{-1}(]-\infty,r[)\) has at least \(k\) connected components in the weak topology, then for each \(\lambda>0\) small enough the functional \(\Psi+\lambda\Phi\) has at least \(k\) local minima in \(\Psi^{-1}(]-\infty,r[)\).
The result can be used to provide information on the number of the local minima of suitable perturbations of \(\Psi\), or to describe the structure of the set of all global minima and of the sublevel sets of \(\Psi\). This last application is developed in the paper when \(\Psi\) is the energy functional related to a Dirichlet problem.

49J27 Existence theories for problems in abstract spaces
49J45 Methods involving semicontinuity and convergence; relaxation
49K40 Sensitivity, stability, well-posedness
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