Quantitative deformation theorems and critical point theory

Abstract

In the framework of critical point theory for continuous functionals defined on metric spaces, we show how quantitative deformation properties can be used to obtain saddle-point type results, even in the case when the usual geometric assumptions are not satisfied. We thus unify and extend to a nonsmooth setting some recent results of Schechter.

A weighted eigenvalue problem for the p-Laplacian plus a potential

“…From Theorem 1 we know that µ(λ) > −∞ is the unique principal eigenvalue of problem (9). Let us denote here by ϕ λ the L p -normalized positive eigenfunction associated to µ(λ).…”

“…A sequence of eigenvalues of (P) by the Ljusternik-Schnirelmann principle can be obtained when either α(V, m) > 0 or α(V, m) = 0 and |Ω − | > 0. To this aim, one may apply, for instance, an equivariant version of a deformation theorem suitable for the (PSC) condition (see [9] or [10]). …”

Brezis–Nirenberg type theorems and multiplicity of positive solutions for a singular elliptic problem

“…There is a vast literature on nonsmooth analysis. To mention just the results we need in our problem, we refer the reader to [5,[9][10][11]18,27]; see also the book [17] which treats nonsmooth analysis in several chapters. The main nonsmooth tool we use is a linking theorem (see Theorem 2) for functions which are a C 1 -perturbation of a convex function; to be precise, we just use the mountain pass version of the above mentioned theorem.…”

Origin and evolution of the Palais–Smale condition in critical point theory

“…A quantitative deformation lemma approach has been used in 1999 by Corvellec [34] to prove various type of minimax theorems extending some results of Schechter mentioned above to the nonsmooth frame based upon the weak slope.…”