Differential Systems with Strongly Indefinite Variational Structure

Strongly indefinite systems with critical Sobolev exponents

“…This is shown in [HV1] and [FF] by means of an indefinite functional theorem due to Benci and Rabinowitz [BR], and in [CFM1] and [PV] by topological methods. Besides a PS-condition, this theorem requires compactness of the gradient of the non-quadratic part of the above Lagrangian, which fails in the critical case (4).…”

“…As in [HV1] and [FF], weak solutions of (I p,q ) correspond to critical points of the functional f defined on a suitable member of a family of Sobolev interpolation spaces. For a precise description of these spaces in terms of Fourier series, see [HV1], [FF].…”

“…Alternatively, by (linear) regularity theory (Agmon-Douglis-Nirenberg estimates; see [HV1] again), we can also view it as an isomorphism…”

“…is a critical point of f : E α → R, regularity theory (see [HV1]) implies that A λ,µ z ∈ X * α . Consequently we may obtain a critical point of f * : X * α → R by setting w = i * w = A λ,µ z.…”

“…One sees that the coupling now also occurs in the quadratic part of f , and therefore it is much stronger than in the potential case. An immediate consequence is that there is no longer only one appropriate choice of function spaces [HV1], [FF], because there is a trade-off between u and v. This accounts for a much richer structure and a rather different characterization of the concept of critical growth. Consider for example…”