On a fourth-order quasilinear elliptic equation of concave–convex type

Abstract

Abstract.We consider a fourth-order quasilinear equation depending on a nonnegative parameter λ and with subcritical or critical growth. Such equation is equivalent to a Hamiltonian system and the main goal of this work is to prove the existence of at least two positive and infinitely many solutions for such equation when the parameter λ is positive and small enough. Mathematics Subject Classification (2000). Primary 35J40; Secondary 35J55.

Representation theorems for Sobolev spaces on intervals and multiplicity results for nonlinear ODEs

“…It turns out that the weak solutions for (5.1) produce classical solutions for (5.2); see [11,Theorem 1.1]. Theorem 1.3 in [11] restricted to the one-dimensional case, see also some comments in [11, Section 1], states the following results concerning (5.1).…”

“…In particular, we obtain some extension in the one-dimensional case of the results in dos Santos [11].…”

“…When working with equations involving the polyharmonic (m-harmonic) operator under Navier boundary conditions (see [19] and [9][10][11]), the following topological isomorphisms are useful:…”

“…In some cases, it turns out to be important to know a concrete Schauder basis, that is, to present explicitly the elements of such Schauder basis. See for instance [5,11,15,29].…”

“…As presented in [9,11], the appropriated space to treat (5.1) is the Sobolev space E p introduced at Definition 3.1. We define the weak solutions for (5.1) as the critical points of…”