Existence Results for Super-Liouville Equations on the Sphere via Bifurcation Theory

Abstract

We are concerned with super-Liouville equations on S 2 , which have variational structure with a strongly-indefinite functional. We prove the existence of nontrivial solutions by combining the use of Nehari manifolds, balancing conditions and bifurcation theory.

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

“…Locally, N ρ is the negative space of the Hessian Hess(J ρ ) at (0, 0). In principle, since the dimension of N ρ changes when ρ runs across an eigenvalue, this would imply that there is a bifurcation phenomena occurring when ρ is close to the eigenvalues of / D g , as discussed in [23]. Here we will show that there are solutions for all ρ ∈ (0, ∞) \ Spect( / D g ).…”

“…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.…”

“…see [1,35], where W = W (v) is a superpotential, v is a scalar function and / D is the Dirac operator acting on spinors φ, see Section 2 for precise definitions. For instance, the super Liouville equations, which have attracted a great attention [2,15,22,23,26,27,28,29,36], are recovered by considering the potential W (v) = 8πµe bv . Here we will be concerned with the super sinh-Gordon equations (SShG) with potential W (v) = 8πµ cosh(bv), (1) where µ > 0 and b > 0 are physical parameters, see [1,35].…”

Conformal Dirac-Einstein equations on manifolds with boundary

“…Observe that the functional (1) can be considered as the threedimensional analogue of the Super-Liouville functional, for which a blow-up analysis and the existence of critical points has been studied in the literature, see e.g. [27,23,24] (for the case ∂M = ∅) and [13,28,29,30] (for the case ∂M = 0).…”

“…loc (ΣR 3 + ) and strongly in L p loc (ΣR 3 + ) for p < 3. Now we notice that from (24), we have that…”