# zbMATH — the first resource for mathematics

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.

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