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

Bonheure, Denis; Santos, Ederson Moreira dos

doi:10.1016/j.jde.2010.08.014

Cited by 2 publications

(1 citation statement)

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“…Indeed, for instance, if N = 2 and Ω = (0, 1) × (0, 1), the sequence of eigenfunctions of (−∆, H 1 0 (Ω)), ordered according to the corresponding increasing value of the sequence of eigenvalues, is not a Schauder base for L (p+1)/p (Ω) if p = 1, since the process of "ball summation" for the double Fourier series does not work; see [71, Section 3.3 & Theorem 3.5.6]. We refer to [26] for the complete proof of Theorem 8.5.…”

Section: The Symmetric Mountain Pass Lemma Combined With the Reductiomentioning

confidence: 99%

Hamiltonian elliptic systems: a guide to variational frameworks

Bonheure¹,

Santos²,

Tavares³

2014

Port. Math.

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Abstract. Consider a Hamiltonian elliptic system of type    −∆u = Hv (u, v) in Ω −∆v = Hu (u, v) in Ω u, v = 0 on ∂Ω where H is a power-type nonlinearity, for instancehaving subcritical growth, and Ω is a bounded domain of R N , N ≥ 1. The aim of this paper is to give an overview of the several variational frameworks that can be used to treat such a system. Within each approach, we address existence of solutions, and in particular of ground state solutions. Some of the available frameworks are more adequate to derive certain qualitative properties; we illustrate this in the second half of this survey, where we also review some of the most recent literature dealing mainly with symmetry, concentration, and multiplicity results. This paper contains some original results as well as new proofs and approaches to known facts.

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Section: The Symmetric Mountain Pass Lemma Combined With the Reductiomentioning

confidence: 99%