Concentration estimates for Emden-Fowler equations with boundary singularities and critical growth

Abstract

We establish -among other things-existence and multiplicity of solutions for the Dirichlet problem i ∂ ii u+ |u| 2 ⋆ −2 u |x| s = 0 on smooth bounded domains Ω of R n (n ≥ 3) involving the critical Hardy-Sobolev exponent 2 ⋆ = 2(n−s) n−2 where 0 < s < 2, and in the case where zero (the point of singularity) is on the boundary ∂Ω. Just as in the Yamabe-type non-singular framework (i.e., when s = 0), there is no nontrivial solution under global convexity assumption (e.g., when Ω is star-shaped around 0). However, in contrast to the nonsatisfactory situation of the non-singular case, we show the existence of an infinite number of solutions under an assumption of local strict concavity of ∂Ω at 0 in at least one direction. More precisely, we need the principal curvatures of ∂Ω at 0 to be non-positive but not all vanishing. We also show that the best constant in the Hardy-Sobolev inequality is attained as long as the mean curvature of ∂Ω at 0 is negative, extending the results of [21] and completing our result of [22] to include dimension 3. The key ingredients in our proof are refined concentration estimates which yield compactness for certain Palais-Smale sequences which do not hold in the non-singular case.1 1 In our context, we specify the orientation of ∂Ω in such a way that the normal vectors of ∂Ω are pointing outward from the domain Ω.

Sobolev inequalities for the Hardy–Schrödinger operator: extremals and critical dimensions

“…• The other approach was initiated by Ghoussoub and Kang [25] and developed by Ghoussoub and Robert [28][29][30] when s > 0 and γ = 0, and by C.S. Lin et al [37,[45][46][47] and Ghoussoub and Robert [31] when γ = 0.…”

“…For s = γ = 0, the general pointwise estimates are performed in the monograph [21] of Druet-HebeyRobert. We also refer to Ghoussoub and Robert [29] for the optimal control with arbitrary high energy when s > 0 and γ = 0. Other methods developed to get pointwise estimates are due to Schoen and Zhang [54] and Kuhri et al [42].…”

“…The result of Kang-Ghoussoub was eventually improved later by Ghoussoub and Robert [28,29], who also proved it for n = 3 and by only requiring that the mean curvature, i.e., the trace of the second fundamental form, at 0, to be negative (see also Chern and Lin [16]). Qualitatively, this says that there are extremals for μ 0,s ( ), whenever the domain at 0 has more concave directions than convex ones, in the sense that the negative principal directions dominate quantitatively the positive principal directions.…”

“…The proof-due to Ghoussoub and Robert [28]-uses the machinery developed in Druet et al [21] for equations of Yamabe-type on manifolds. It also allows to tackle problems with arbitrary high energy and not just minima [29].…”

The role of the mean curvature in a Hardy–Sobolev trace inequality

“…Actually, by the results in [10], extremals for Q(s, Ω) exists if Ω is "average concave in a neighborhood of the origin". Later on, in the same year, Ghoussoub and Robert [15,16] used refined blow-up analysis to prove existence of an extremal for Q(s, Ω) provided the mean curvature of ∂Ω is negative at 0.…”