Characterization of the Palais–Smale sequences for the conformal Dirac–Einstein problem and applications

Abstract

In this paper we study the Palais-Smale sequences of the conformal Dirac-Einstein problem.After we characterize the bubbling phenomena, we prove an Aubin type result leading to the existence of a positive solution. Then we show the existence of infinitely many solutions to the problem provided that the underlying manifold exhibits certain symmetries.

Bifurcating standing waves for effective equations in gapped honeycomb structures

“…Such types of equations have been studied also in different contexts. We mention, for instance, problems from conformal spin geometry, for which we refer the reader to [12][13][14][15] and references therein, and in the case of coupled systems involving the Dirac operator and critical nonlinearities related to supersymmetric models coupling gravity with fermions, see [16,17]. The main difficulty in studying those equations comes from the underlying conformal symmetry so that looking for stationary solutions by variational methods one has to deal with the induced loss of compactness, see [10,11].…”

Min-max solutions for super sinh-Gordon equations on compact surfaces

“…The difficulty in dealing with such equations is due to the strong indefiniteness of the Dirac operator, and a typical useful strategy is to use some Nehari type manifold to kill most of the negative directions, see e.g. [33,22,23] and also [38,39] for a more general treatment. Here we will adopt the same approach.…”

“…Fortunately we can verify it as long as ρ / ∈ Spect( / D). We are inspired here by some results in [7,33,37].…”

Existence results for the conformal Dirac-Einstein system

“…Regarding the first issue, in [19] the authors studied the lack of compactness and gave a precise description of the bubbling phenomena, characterizing the behaviour of the Palais-Smale sequences, in the spirit of classical works [22,21,23,14,15,5,3]. For the strongly indefinite difficulty, in [16,17,18] general functionals with these features are studied by using methods based on a homological approach.…”