Multiplicity and Stability of the Pohozaev Obstruction for Hardy-Schro center dot dinger Equations with Boundary Singularity

Let 12 be a smooth bounded domain in Rn (n >= 3) such that 0 is an element of a 12. We consider issues of non-existence, existence, and multiplicity of variational solutions in 1-121,0(12) for the borderline Dirichlet problem, { -Delta u - eta u |x|2 - h(x)u = |u|2. (s)-2u |x|s in 12, u = 0 on a12 \ {0}, (E) where 0 < s < 2, 2 ⠂(s) := 2(n-s) n-2 , eta is an element of R and h is an element of C0(12). We use sharp blow-up analysis on-possibly high energy-solutions of corresponding subcritical problems to establish, for example, that if eta < n24 - 1 and the principal curvatures of a12 at 0 are non-positive but not all of them vanishing, then Equation (E) has an infinite number of high energy (possibly sign-changing) solutions in 1-121,0(12). This complements results of the first and third authors, who showed in their 2016 article, Hardy-Singular Boundary Mass and Sobolev-Critical Variational Problems, that if eta <= n24 - 14and the mean curvature of a12 at 0 is negative, then (E) has a positive least energy solution. On the other hand, the sharp blow-up analysis also allows us to show that if the mean curvature at 0 is nonzero and the mass, when defined, is also nonzero, then there is a surprising stability of regimes where there are no variational positive solutions under C1-perturbations of the potential h. In particular, and in sharp contrast with the non-singular case (i.e., when eta = s = 0), we prove non-existence of such solutions for (E) in any dimension, whenever 12 is star-shaped and h is close to 0, which include situations not covered by the classical Pohozaev obstruction.