The Rabinowitz–Floer homology for a class of semilinear problems and applications

Abstract

In this paper, we construct a Rabinowitz-Floer type homology for a class of non-linear problems having a starshaped potential; we consider some equivariant cases as well. We give an explicit computation of the homology and we apply it to obtain results of existence and multiplicity of solutions for several model equations.

New existence results for the mean field equation on compact surfaces via degree theory

“…With suitable nonlinearities as perturbation adding to the geometric equations, T. Isobe made remarkable progress in adapting the classical theory of calculus of variations to the Dirac setting [18,19,20]. Combined with the methods of Robinowitz-Floer homology, A. Maalaoui and V. Martino also obtained existence results of some nonlinear Dirac type equations, see [29,30,31] and the references therein. In the case of super-Liouville equations we have to deal with an exponential nonlinearity, which does not fit in the above settings.…”

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

“…Since the functional is conformally invariant, one expects compactness to be violated for this problem; moreover, due to the presence of the Dirac operator, it is strongly indefinite. For the later part, the authors studied an effective method, based on a homological approach [21,23,24], for general functionals with this feature of being strongly indefinite; here we will focus on the first issue, that is the lack of compactness. We recall that a C 1 function F satisfies the Palais-Smale condition (PS) if: for any sequence x k such that F (x k ) → c and ∇F (x k ) → 0 (such a sequence is then called a (PS) sequence), there exists a converging subsequence.…”

On Coupled Dirac Systems under Boundary Condition

“…A typical way to deal with such problems is the min-max method of Benci and Rabinowitz [10], including the mountain pass theorem, linking arguments. Another is a homological method, Morse theory and Rabinowitz-Floer homology as in [1,9,27,29,30]. For the Dirac operator associated with appropriate boundary condition , Farinell, Schwarz [15] prove that Dirac operator P is elliptic and extends to a self-adjoint operator with a discrete spectrum.…”