The Hardy-Schrödinger operator on the Poincaré ball: compactness, multiplicity, and stability of the Pohozaev obstruction (Q2118896)

Let \(\Omega\) be a smooth relatively compact domain containing zero in the Poincaré model of the hyperbolic space \(\mathbb{B}^n\) with \(n \geq 3\). The authors of the paper under review consider the Dirichlet problem \[ \begin{cases} -\Delta_{\mathrm{B}} u - \gamma V_2 u - \lambda u =V_{2^{\star}(s)} |u|^{2^{\star}(s)-2} u \quad \text{ in } \Omega, \\ u = 0 \text{ on } \partial \Omega, \end{cases} \tag{1} \] where \(\Delta_{\mathrm{B}}\) denotes the Laplace-Beltrami on \(\mathbb{B}^n\) and \(V_2\) (resp. \(V_{2^{\star}(s)}\)) is a Hardy-type potential (resp. Hardy-Sobolev weight) that is invariant under hyperbolic scaling and behaves as \(r^{-2}\) (resp. \(r^{-s}\)) at the origin. Moreover, \(\gamma < (n-2)^2/4\), \(0 < s < 2\) and \(2^{\star}(s) = 2(n-s)/(n-2)\). The authors are interested in the problems of non-existence, existence, and multiplicity of variational solutions of the borderline problem in (1). They prove the existence of positive ground state solutions of (1) for \(n \geq 4\), \(0 \leq \gamma \leq (n-2)^2/4-1\) and \(\lambda > 0\). The statement also holds if \(n \geq 3\) and \(\gamma > (n-2)^2/4-1\) provided the domain has a positive hyperbolic mass. This improves similar results in [\textit{H. Chan} et al., Adv. Nonlinear Stud. 18, No. 4, 671--689 (2018; Zbl 1407.35074)]. If \(\gamma>(n-2)^2/4-1\) and the hyperbolic mass is non-vanishing, the authors identify regimes, where no positive variational solution exists. For \(n \geq 5\), \(\gamma < (n-2^2)/4-4\) and \(\lambda > (n (n-4)/4 - \gamma) (n-2)/(n-4)\) they show that infinitely many higher energy solutions to (1) exist. To prove the above statements the authors apply a conformal change of metric to reduce (1) to a Dirichlet boundary value problem in Euclidean space. Versions of this problem in Euclidean space have been studied extensively in the literature. We refer to [\textit{H. Brézis} and \textit{L. Nirenberg}, Commun. Pure Appl. Math. 36, 437--477 (1983; Zbl 0541.35029); \textit{O. Druet}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 19, No. 2, 125--142 (2002; Zbl 1011.35060); Indiana Univ. Math. J. 51, No. 1, 69--88 (2002; Zbl 1037.58012); \textit{O. Druet} and \textit{P. Laurain}, J. Eur. Math. Soc. (JEMS) 12, No. 5, 1117--1149 (2010; Zbl 1210.35105); \textit{G. Devillanova} and \textit{S. Solimini}, Adv. Differ. Equ. 7, No. 10, 1257--1280 (2002; Zbl 1208.35048)] for the non-singular case (\(s=0=\gamma\)), to [\textit{E. Jannelli}, J. Differ. Equations 156, No. 2, 407--426 (1999; Zbl 0938.35058); \textit{D. Cao} and \textit{S. Yan}, Calc. Var. Partial Differ. Equ. 38, No. 3--4, 471--501 (2010; Zbl 1194.35161); \textit{F. Catrina} and \textit{R. Lavine}, Commun. Contemp. Math. 4, No. 3, 529--545 (2002; Zbl 1013.35023); \textit{P. Esposito} et al., Anal. PDE 14, No. 2, 533--566 (2021; Zbl 1473.35290); the first and the third author, Anal. PDE 10, No. 5, 1017--1079 (2017; Zbl 1379.35077)] for the case \(s=0\), \(\gamma \neq 0\) and to [the first and the third author, Calc. Var. Partial Differ. Equ. 56, No. 5, Paper No. 149, 54 p. (2017; Zbl 1384.35012); \textit{D. Cao} and \textit{S. Peng}, J. Differ. Equations 193, No. 2, 424--434 (2003; Zbl 1140.35412); \textit{S. Peng} and \textit{C. Wang}, Math. Methods Appl. Sci. 38, No. 2, 197--220 (2015; Zbl 1308.35098); \textit{C. Wang} and \textit{J. Wang}, Commun. Contemp. Math. 14, No. 6, 1250044, 38 p. (2012; Zbl 1310.35117); \textit{S. Yan} and \textit{J. Yang}, Calc. Var. Partial Differ. Equ. 48, No. 3--4, 587--610 (2013; Zbl 1280.35048); the first and the third author, Anal. PDE 10, No. 5, 1017--1079 (2017; Zbl 1379.35077); the first author et al., ``Multiplicity and stability of the Pohozaev obstruction for Hardy-Schrödinger equations with boundary singularity'', in: Memoirs of the AMS, in press] for the case when both, \(s\) and \(\gamma\), are non-zero.