Infinitely many solutions of some nonlinear variational equations

Abstract

The aim of this paper is investigating the existence of one or more critical points of a family of functionals which generalizes the model problem \ud \[\bar J(u) = \int_\Omega \bar A(x,u)|\nabla u|^p dx - \int_\Omega G(x,u) dx\]\ud in the Banach space $W^{1,p}_0(\Omega)\cap L^\infty(\Omega)$, being $\Omega$ a bounded domain in $\R^N$.\ud In order to use ``classical'' theorems, a suitable variant of condition $(C)$ is proved and $W^{1,p}_0(\Omega)$ is decomposed according to a ``good'' sequence of finite dimensional subspaces

Existence of entire solutions for a class of quasilinear elliptic equations

“…We start by establishing a geometrical property for the energy functional Φ λ , which is valid for all λ > 0 and is a variant of the geometrical structure of the Mountain Pass theorem due to Ambrosetti and Rabinowitz [4]. For a similar result, obtained with a different proof and the use of the Palais-Smale compactness condition, we refer to [7].…”

“…The proof of Theorem A.3 is based on the Ekeland variational principle, see for instance [16]. For a similar generalization of the Mountain Pass theorem, with a different proof and the use of a compactness condition, we refer to Theorem 2.5 of [7]. Theorem A.3.…”

p-Laplacian problems with nonlinearities interacting with the spectrum

“…[17]). With respect to the semilinear case, the drawback in using eigenvalues is that, for the Banach space W As we shall see in the proofs of our main results, the definition of the quasi-eigenvalues proposed in [10] fits in with our purposes as a suitable decomposition of the Sobolev space W 1,p 0 (Ω) can be introduced, so that it turns out to be the classical one for p = 2.…”

“…On the other hand, when p = 2 the full spectrum of −Δ p is still unknown, even if various authors have been introducing different characterizations of eigenvalues and definitions of quasi-eigenvalues. Here, we use two sequences of quasi-eigenvalues, denoted by (η k ) k and (ν k ) k , following respectively [10,16] (such sequences and their properties are fully described in Sect. 2).…”

Multiple solutions for some symmetric supercritical problems

“…In the past, such a problem has been overcome by introducing suitable definitions of critical point for J and related existence results have been stated (see, e.g., [2,3,11,15]). Here, as in [7], suitable assumptions assure that the functional J is C 1 in X = W 1,p 0 (Ω) ∩ L ∞ (Ω) (see Proposition 3.2) and its Euler-Lagrange equation is −div(a(x, u, ∇u)) + A t (x, u, ∇u) = g(x, u) in Ω, u = 0 on ∂Ω,…”