Nonsmooth critical point theory and quasilinear elliptic equations

Quantitative deformation theorems and critical point theory

“…See [4] and the references therein for further comparisons between the weak slope and various analytical notions involving functionals which are not, in general, (Gâteaux) differentiable. Definition 2.2.…”

“…The results of this paper are developed in view of applications to nonsmooth variational problems, such as quasilinear elliptic problems of the type studied in [3,4] (see also the references in [4]). …”

Positive solutions of critical quasilinear elliptic problems ingeneral domains

“…Therefore, as and (1) is solved in a weak sense. We refer to the original papers [9,14,15] for the basic definitions in this nonsmooth context; this theory has been widely used for different problems related to quasilinear elliptic equations of the kind of (1), see [3,4,9,10,13]. Under suitable assumptions on the functions a ij , g and p, in this paper we prove the existence of positive solutions of (1) in bounded and unbounded domains Ω: making use of the techniques introduced in [21], we prove our results for a wide class of subcritical perturbations g. As far as we are aware, very few results concerning (1) are known: apart the already mentioned case with p(x) ≡ 1 on bounded domains [4], we refer to [24] where a minimization problem related to (1) is solved for Ω = IR n and to [27] for a similar problem in a bounded domain.…”

A Variant of Clark's Theorem and Its Applications for Nonsmooth Functionals without the Palais--Smale Condition

“…Choose H ∈ C 1 (R) such that H(s) = 1 for |s| ≤ 1/2, H(s) = 0 for |s| ≥ 1, 0 ≤ H(s) ≤ 1, and |H (s)| ≤ 4 for all s. Let ψ ∈ E with ψ ≥ 0 and let T ∈ R with T ≥ 1. As in [4], take…”

“…This theory has many important applications; see, for example, [4,14,15]. In particular, in [15] Degiovanni and Schuricht obtained a Clark-type theorem for lower semicontinuous functionals defined in metric spaces (see [15,Theorem 2.5]).…”