Asymptotic behaviour of ground states

Abstract

Abstract. We derive the asymptotic behaviour of the ground states of a system of two coupled semilinear Poisson equations with a strongly indefinite variational structure in the critical Sobolev growth case.

Ground state and non-ground state solutions of some strongly coupled elliptic systems

“…The idea of such a reduction goes back at least to P.-L. Lions [24]; see also [38,23,10,11,18]. It turns out that (H1), the hypothesis for subcriticality for (1.1), is…”

Strongly indefinite systems with critical Sobolev exponents

“…The main advantage is now that critical points of the dual functional of (6), despite its original indefinite character, can be obtained as mountain passes, provided that the corresponding critical value lies below a constant depending only on p, q and n, which is related to the best Sobolev constant appearing in a naturally associated Sobolev inequality. To get below this constant the exact asymptotic behaviour of the regular radial ground states of system (I p,q ) with λ = µ = 0 and Ω = R n as established in [HV2] is required.…”

“…In section 3 we discuss the geometric properties of the dual functional. Finally, in section 4, we use the asymptotic estimates proved in [HV2] of the ground states for (I p,q ) with λ = µ = 0 in R n , to push the energy level below the critical Palais-Smale level. This allows us to complete the proof of Theorem 2.…”

“…Note that the limit w is a stationary point of f * and that its critical value equals c. In order to see for which values of c (if any) the function f * satisfies (P S) c , let us introduce (see also [BrN], [L1], [L2], [HV2]) the embedding constants…”

“…The existence of a critical point of mountain pass type will follow from a standard min-max argument. For this purpose we need the precise decay rates [HV2] of the ground states of (I p,q ) with λ = µ = 0 and (p, q) on the critical hyperbola. Let us denote this one-parameter family of ground states by z = (u , v ), where ∈ (0, ∞).…”