Abstract. In this paper we investigate the existence of solutions to a nonlinear elliptic problem involving critical Sobolev exponent for a polyharmonic operator on a Riemannian manifold M . We first show that the best constant of the Sobolev embedding on a manifold can be chosen as close as one wants to the Euclidean one, and as a consequence derive the existence of minimizers when the energy functional goes below a quantified threshold. Next, higher energy solutions are obtained by Coron's topological method, provided that the minimizing solution does not exist. To perform this topological argument, we overcome the difficulty of dealing with polyharmonic operators on a Riemannian manifold and adapting Lions's concentration-compactness lemma. Unlike Coron's original argument for a bounded domain in R n , we need to do more than chopping out a small ball from the manifold M . Indeed, our topological assumption that a small sphere on M centred at a point p ∈ M does not retract to a point in M \{p} is necessary, as shown for the case of the canonical sphere where chopping out a small ball is not enough.
We study the Hardy identities and inequalities on Cartan-Hadamard manifolds using the notion of a Bessel pair. These Hardy identities offer significantly more information on the existence/nonexistence of the extremal functions of the Hardy inequalities. These Hardy inequalities are in the spirit of Brezis-Vázquez in the Euclidean spaces. As direct consequences, we establish several Hardy type inequalities that provide substantial improvements as well as simple understandings to many known Hardy inequalities and Hardy-Poincaré-Sobolev type inequalities on hyperbolic spaces in the literature.
Given a high-order elliptic operator on a compact manifold with or without boundary, we perform the decomposition of Palais-Smale sequences for a nonlinear problem as a sum of bubbles. This is a generalization of the celebrated 1984 result of Struwe [16]. Unlike the case of second-order operators, bubbles close to the boundary might appear. Our result includes the case of a smooth bounded domain of R n .
We consider the Hardy-Schrödinger operator Lγ := −∆ B n − γV 2 on the Poincaré ball model of the Hyperbolic space B n (n ≥ 3). Here V 2 is a well chosen radially symmetric potential, which behaves like the Hardy potential around its singularity at 0, i.e., V 2 (r) ∼ 1 r 2 . Just like in the Euclidean setting, the operator Lγ is positive definite whenever γ < (n−2) 2 4 , in which case we exhibit explicit solutions for the Sobolev critical equation Lγ u = V 2 * (s) u 2 * (s)−1 in B n , where 0 ≤ s < 2, 2 * (s) = 2(n−s) n−2 , and V 2 * (s) is a weight that behaves like 1 r s around 0. In dimensions n ≥ 5, the above equation in a domain Ω of B n containing 0 and away from the boundary, has a ground state solution, whenever 0 < γ ≤ n(n−4) 4 , and provided Lγ is replaced by a linear perturbation Lγ − λu, where λ > n − 2 n − 4 n(n − 4) 4 − γ . On the other hand, in dimensions 3and 4, the existence of solutions depends on whether the domain has a postive "hyperbolic mass," a notion that we introduce and analyze therein.
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