Weak solutions of quasilinear elliptic PDE's at resonance

Nontrivial Solutions of Quasilinear Equations via Nonsmooth Morse Theory

“…We want to give a result of the afore-mentioned type for the quasilinear elliptic problem where again g(x, 0)=0. Only few existence results have been proved so far, for this kind of equations, by means of techniques of critical point theory; we can mention [18], and the recent papers [2,3,5,6]. This is due to the fact that classical critical point theory is not fit for quasilinear problems of this type.…”

“…About the mentioned papers, we also observe that nonsmooth critical point theory is involved through techniques of Lusternik Schnirelman or mountain pass type. In [3,5,6], in particular, the abstract theory of [13,11] is exploited.…”

Quantitative deformation theorems and critical point theory

“…In the case when X is a Banach space,Ã is the origin, and β(t) = 1/(1 + t), the metricd defined in the above proof is the Cerami metric, see [5], [10, p. 138], where it is used in the context of critical point theory for "smooth" functions. This metric is used in [1] in the context of the critical point theory for continuous functions defined on complete metric spaces developed in [9,8]. Since (X,d) is complete (see Theorem 4.1 (a)), and the Palais-Smale condition "with weight" β( u ) −1 = 1 + u (see, e.g., [16] for the terminology) is just the condition of Definition 2.2 in (X,d), existence results of critical points of a continuous function f : X → R readily follow from the general theory (see Theorem 3.3 (a)).…”

Positive solutions of critical quasilinear elliptic problems ingeneral domains

“…Under reasonable assumptions on a ij , g, p, the functional J is continuous but not even locally Lipschitz if the functions a ij (x, s) depend on s, see [9]. However, J is weakly C ∞ c (Ω)-differentiable (see [3,9]) and the derivative of J exists in the smooth directions: for all u ∈ D according to the nonsmooth critical point theory of [14,15] it is possible to prove that critical points u of J satisfy J (u)[ϕ] = 0 ∀ϕ ∈ C ∞ c (Ω) and hence solve (1) in distributional sense, see also [2]. Therefore, as and (1) is solved in a weak sense.…”

“…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.…”