# American Institute of Mathematical Sciences

May  2015, 14(3): 1097-1125. doi: 10.3934/cpaa.2015.14.1097

## Klein-Gordon-Maxwell equations in high dimensions

 1 Université de Cergy-Pontoise, CNRS, Département de Mathématiques, F-95000 Cergy-Pontoise, France

Published  March 2015

We prove the existence of a mountain-pass solution and the a priori bound property for the electrostatic Klein-Gordon-Maxwell equations in high dimensions.
Citation: Pierre-Damien Thizy. Klein-Gordon-Maxwell equations in high dimensions. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1097-1125. doi: 10.3934/cpaa.2015.14.1097
##### References:
 [1] Antonio Ambrosetti and Paul H. Rabinowitz, Dual variational methods in critical point theory and applications,, \emph{J. Functional Analysis}, 14 (1973), 349. Google Scholar [2] Thierry Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire,, \emph{J. Math. Pures Appl.}, 55 (1976), 269. Google Scholar [3] A. Azzollini, L. Pisani and A. Pomponio, Improved estimates and a limit case for the electrostatic Klein-Gordon-Maxwell system,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 141 (2011), 449. doi: 10.1017/S0308210509001814. Google Scholar [4] Antonio Azzollini and Alessio Pomponio, Ground state solutions for the nonlinear Klein-Gordon-Maxwell equations,, \emph{Topol. Methods Nonlinear Anal.}, 35 (2010), 33. Google Scholar [5] Vieri Benci and Claudio Bonanno, Solitary waves and vortices in non-Abelian gauge theories with matter,, \emph{Adv. Nonlinear Stud.}, 12 (2012), 717. Google Scholar [6] Vieri Benci and Donato Fortunato, Solitary waves of the nonlinear Klein-Gordon equation coupled with the Maxwell equations,, \emph{Rev. Math. Phys.}, 14 (2002), 409. doi: 10.1142/S0129055X02001168. Google Scholar [7] Vieri Benci and Donato Fortunato, Solitary waves in the nonlinear wave equation and in gauge theories,, \emph{J. Fixed Point Theory Appl.}, 1 (2007), 61. doi: 10.1007/s11784-006-0008-z. Google Scholar [8] Vieri Benci and Donato Fortunato, Existence of hylomorphic solitary waves in Klein-Gordon and in Klein-Gordon-Maxwell equations,, \emph{Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl.}, 20 (2009), 243. doi: 10.4171/RLM/546. Google Scholar [9] Vieri Benci and Donato Fortunato, Spinning $Q$-balls for the Klein-Gordon-Maxwell equations,, \emph{Comm. Math. Phys.}, 295 (2010), 639. doi: 10.1007/s00220-010-0985-z. Google Scholar [10] Haïm Brézis and Louis Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, \emph{Comm. Pure Appl. Math.}, 36 (1983), 437. doi: 10.1002/cpa.3160360405. Google Scholar [11] Luis A. Caffarelli, Basilis Gidas and Joel Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, \emph{Comm. Pure Appl. Math.}, 42 (1989), 271. doi: 10.1002/cpa.3160420304. Google Scholar [12] Daniele Cassani, Existence and non-existence of solitary waves for the critical Klein-Gordon equation coupled with Maxwell's equations,, \emph{Nonlinear Anal.}, 58 (2004), 733. doi: 10.1016/j.na.2003.05.001. Google Scholar [13] Monica Clapp, Marco Ghimenti and Anna Maria Micheletti, Semiclassical states for a static supercritical Klein-Gordon-Maxwell-Proca system on a closed Riemannian manifold,, Preprint, (2013). Google Scholar [14] Olivier Druet, Compactness for Yamabe metrics in low dimensions,, \emph{Int. Math. Res. Not.}, 23 (2004), 1143. doi: 10.1155/S1073792804133278. Google Scholar [15] Olivier Druet and Emmanuel Hebey, Stability for strongly coupled critical elliptic systems in a fully inhomogeneous medium,, \emph{Anal. PDE}, 2 (2009), 305. doi: 10.2140/apde.2009.2.305. Google Scholar [16] Olivier Druet and Emmanuel Hebey, Existence and a priori bounds for electrostatic Klein-Gordon-Maxwell systems in fully inhomogeneous spaces,, \emph{Commun. Contemp. Math.}, 12 (2010), 831. doi: 10.1142/S0219199710004007. Google Scholar [17] Olivier Druet, Emmanuel Hebey and Paul Laurain, Stability of elliptic PDEs with respect to perturbations of the domain,, \emph{J. Differential Equations}, 255 (2013), 3703. doi: 10.1016/j.jde.2013.07.051. Google Scholar [18] Olivier Druet, Emmanuel Hebey and Frédéric Robert, Blow-up Theory for Elliptic PDEs in Riemannian Geometry,, Mathematical Notes, 45 (2004). doi: 10.1007/BF01158557. Google Scholar [19] Olivier Druet, Emmanuel Hebey and Jérôme Vétois, Bounded stability for strongly coupled critical elliptic systems below the geometric threshold of the conformal Laplacian,, \emph{J. Funct. Anal.}, 258 (2010), 999. doi: 10.1016/j.jfa.2009.07.004. Google Scholar [20] Olivier Druet, Emmanuel Hebey and Jérôme Vétois, Stable phases for the 4-dimensional KGMP system.,, \emph{J. Reine Angew. Math.}, (). Google Scholar [21] Olivier Druet and Paul Laurain, Stability of the Pohožaev obstruction in dimension 3,, \emph{J. Eur. Math. Soc. (JEMS)}, 12 (2010), 1117. doi: 10.4171/JEMS/225. Google Scholar [22] Teresa D'Aprile and Dimitri Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations,, \emph{Adv. Nonlinear Stud.}, 4 (2004), 307. Google Scholar [23] Teresa D'Aprile and Dimitri Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 134 (2004), 893. doi: 10.1017/S030821050000353X. Google Scholar [24] Teresa D'Aprile and Juncheng Wei, Layered solutions for a semilinear elliptic system in a ball,, \emph{J. Differential Equations}, 226 (2006), 269. doi: 10.1016/j.jde.2005.12.009. Google Scholar [25] Teresa D'Aprile and Juncheng Wei, Solutions en grappe autour des centres harmoniques d'un système elliptique couplé,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 24 (2007), 605. doi: 10.1016/j.anihpc.2006.04.003. Google Scholar [26] Pietro d'Avenia and Lorenzo Pisani, Nonlinear Klein-Gordon equations coupled with Born-Infeld type equations,, \emph{Electron. J. Differential Equations}, (2002). Google Scholar [27] P. d'Avenia, L. Pisani and G. Siciliano, Dirichlet and Neumann problems for Klein-Gordon-Maxwell systems,, \emph{Nonlinear Anal.}, 71 (2009). doi: 10.1016/j.na.2009.02.111. Google Scholar [28] Pietro d'Avenia, Lorenzo Pisani and Gaetano Siciliano, Klein-Gordon-Maxwell systems in a bounded domain,, \emph{Discrete Contin. Dyn. Syst.}, 26 (2010), 135. doi: 10.3934/dcds.2010.26.135. Google Scholar [29] Pierpaolo Esposito, Angela Pistoia and Jérôme Vétois, The effect of linear perturbations on the Yamabe problem,, \emph{Math. Ann.}, (). doi: 10.1007/s00208-013-0971-9. Google Scholar [30] Vladimir Georgiev and Nicola Visciglia, Solitary waves for Klein-Gordon-Maxwell system with external Coulomb potential,, \emph{J. Math. Pures Appl.}, 84 (2005), 957. doi: 10.1016/j.matpur.2004.09.016. Google Scholar [31] Marco Ghimenti and Anna Maria Micheletti, Number and profile of low energy solutions for singularly perturbed Klein-Gordon-Maxwell systems on a Riemannian manifold,, Preprint, (2013). doi: 10.1016/j.jde.2014.01.012. Google Scholar [32] B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations,, \emph{Comm. Partial Differential Equations}, 6 (1981), 883. doi: 10.1080/03605308108820196. Google Scholar [33] David Gilbarg and Neil S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Classics in Mathematics, (2001). Google Scholar [34] Emmanuel Hebey, Solitary waves in critical abelian gauge theories,, \emph{Discrete Contin. Dyn. Syst.}, 32 (2012), 1747. doi: 10.3934/dcds.2012.32.1747. Google Scholar [35] Emmanuel Hebey, Compactness and Stability for Nonlinear Elliptic Equations,, European Mathematical Society, (). doi: 10.4171/134. Google Scholar [36] Emmanuel Hebey and Juncheng Wei, Resonant states for the static Klein-Gordon-Maxwell-Proca system,, \emph{Math. Res. Lett.}, 19 (2012), 953. doi: 10.4310/MRL.2012.v19.n4.a18. Google Scholar [37] Emmanuel Hebey and Juncheng Wei, Schrödinger-Poisson systems in the 3-sphere,, \emph{Calc. Var. Partial Differential Equations}, 47 (2013), 25. doi: 10.1007/s00526-012-0509-0. Google Scholar [38] Emmanuel Hebey and Trong Tuong Truong, Static Klein-Gordon-Maxwell-Proca systems in 4-dimensional closed manifolds,, \emph{J. Reine Angew. Math., 667 (2012), 221. Google Scholar [39] Emmanuel Hebey and Michel Vaugon, The best constant problem in the Sobolev embedding theorem for complete Riemannian manifolds,, \emph{Duke Math. J.}, 79 (1995), 235. doi: 10.1215/S0012-7094-95-07906-X. Google Scholar [40] YanYan Li and Lei Zhang, Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations,, \emph{J. Anal. Math.}, 90 (2003), 27. doi: 10.1007/BF02786551. Google Scholar [41] YanYan Li and Lei Zhang, A Harnack type inequality for the Yamabe equation in low dimensions,, \emph{Calc. Var. Partial Differential Equations}, 20 (2004), 133. doi: 10.1007/s00526-003-0224-y. Google Scholar [42] YanYan Li and Lei Zhang, Compactness of solutions to the Yamabe problem. II,, \emph{Calc. Var. Partial Differential Equations}, 24 (2005), 185. doi: 10.1007/s00526-004-0320-7. Google Scholar [43] Yanyan Li and Meijun Zhu, Yamabe type equations on three-dimensional Riemannian manifolds,, \emph{Commun. Contemp. Math.}, 1 (1999), 1. doi: 10.1142/S021919979900002X. Google Scholar [44] Dimitri Mugnai, Coupled Klein-Gordon and Born-Infeld-type equations: looking for solitary waves,, \emph{Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.}, 460 (2004), 1519. doi: 10.1098/rspa.2003.1267. Google Scholar [45] Dimitri Mugnai, Solitary waves in abelian gauge theories with strongly nonlinear potentials,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 27 (2010), 1055. doi: 10.1016/j.anihpc.2010.02.001. Google Scholar [46] Frédérique Robert and Jérôme Vétois, Examples of non-isolated blow-up for perturbations of the scalar curvature equation,, Preprint, (2012). Google Scholar [47] Frédérique Robert and Jérôme Vétois, A General Theorem for the Construction of Blowing-up Solutions to Some Elliptic Nonlinear Equations with Lyapunov-Schmidt's Finite-dimensional Reduction, Cocompact Imbeddings, Profile Decompositions, and their Applications to PDE,, Trends Math., (2013), 85. Google Scholar [48] Michael Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities,, \emph{Math. Z.}, 187 (1984), 511. doi: 10.1007/BF01174186. Google Scholar [49] Neil S. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds,, \emph{Ann. Scuola Norm. Sup. Pisa}, 22 (1968), 265. Google Scholar

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##### References:
 [1] Antonio Ambrosetti and Paul H. Rabinowitz, Dual variational methods in critical point theory and applications,, \emph{J. Functional Analysis}, 14 (1973), 349. Google Scholar [2] Thierry Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire,, \emph{J. Math. Pures Appl.}, 55 (1976), 269. Google Scholar [3] A. Azzollini, L. Pisani and A. Pomponio, Improved estimates and a limit case for the electrostatic Klein-Gordon-Maxwell system,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 141 (2011), 449. doi: 10.1017/S0308210509001814. Google Scholar [4] Antonio Azzollini and Alessio Pomponio, Ground state solutions for the nonlinear Klein-Gordon-Maxwell equations,, \emph{Topol. Methods Nonlinear Anal.}, 35 (2010), 33. Google Scholar [5] Vieri Benci and Claudio Bonanno, Solitary waves and vortices in non-Abelian gauge theories with matter,, \emph{Adv. Nonlinear Stud.}, 12 (2012), 717. Google Scholar [6] Vieri Benci and Donato Fortunato, Solitary waves of the nonlinear Klein-Gordon equation coupled with the Maxwell equations,, \emph{Rev. Math. Phys.}, 14 (2002), 409. doi: 10.1142/S0129055X02001168. Google Scholar [7] Vieri Benci and Donato Fortunato, Solitary waves in the nonlinear wave equation and in gauge theories,, \emph{J. Fixed Point Theory Appl.}, 1 (2007), 61. doi: 10.1007/s11784-006-0008-z. Google Scholar [8] Vieri Benci and Donato Fortunato, Existence of hylomorphic solitary waves in Klein-Gordon and in Klein-Gordon-Maxwell equations,, \emph{Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl.}, 20 (2009), 243. doi: 10.4171/RLM/546. Google Scholar [9] Vieri Benci and Donato Fortunato, Spinning $Q$-balls for the Klein-Gordon-Maxwell equations,, \emph{Comm. Math. Phys.}, 295 (2010), 639. doi: 10.1007/s00220-010-0985-z. Google Scholar [10] Haïm Brézis and Louis Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, \emph{Comm. Pure Appl. Math.}, 36 (1983), 437. doi: 10.1002/cpa.3160360405. Google Scholar [11] Luis A. Caffarelli, Basilis Gidas and Joel Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, \emph{Comm. Pure Appl. Math.}, 42 (1989), 271. doi: 10.1002/cpa.3160420304. Google Scholar [12] Daniele Cassani, Existence and non-existence of solitary waves for the critical Klein-Gordon equation coupled with Maxwell's equations,, \emph{Nonlinear Anal.}, 58 (2004), 733. doi: 10.1016/j.na.2003.05.001. Google Scholar [13] Monica Clapp, Marco Ghimenti and Anna Maria Micheletti, Semiclassical states for a static supercritical Klein-Gordon-Maxwell-Proca system on a closed Riemannian manifold,, Preprint, (2013). Google Scholar [14] Olivier Druet, Compactness for Yamabe metrics in low dimensions,, \emph{Int. Math. Res. Not.}, 23 (2004), 1143. doi: 10.1155/S1073792804133278. Google Scholar [15] Olivier Druet and Emmanuel Hebey, Stability for strongly coupled critical elliptic systems in a fully inhomogeneous medium,, \emph{Anal. PDE}, 2 (2009), 305. doi: 10.2140/apde.2009.2.305. Google Scholar [16] Olivier Druet and Emmanuel Hebey, Existence and a priori bounds for electrostatic Klein-Gordon-Maxwell systems in fully inhomogeneous spaces,, \emph{Commun. Contemp. Math.}, 12 (2010), 831. doi: 10.1142/S0219199710004007. Google Scholar [17] Olivier Druet, Emmanuel Hebey and Paul Laurain, Stability of elliptic PDEs with respect to perturbations of the domain,, \emph{J. Differential Equations}, 255 (2013), 3703. doi: 10.1016/j.jde.2013.07.051. Google Scholar [18] Olivier Druet, Emmanuel Hebey and Frédéric Robert, Blow-up Theory for Elliptic PDEs in Riemannian Geometry,, Mathematical Notes, 45 (2004). doi: 10.1007/BF01158557. Google Scholar [19] Olivier Druet, Emmanuel Hebey and Jérôme Vétois, Bounded stability for strongly coupled critical elliptic systems below the geometric threshold of the conformal Laplacian,, \emph{J. Funct. Anal.}, 258 (2010), 999. doi: 10.1016/j.jfa.2009.07.004. Google Scholar [20] Olivier Druet, Emmanuel Hebey and Jérôme Vétois, Stable phases for the 4-dimensional KGMP system.,, \emph{J. Reine Angew. Math.}, (). Google Scholar [21] Olivier Druet and Paul Laurain, Stability of the Pohožaev obstruction in dimension 3,, \emph{J. Eur. Math. Soc. (JEMS)}, 12 (2010), 1117. doi: 10.4171/JEMS/225. Google Scholar [22] Teresa D'Aprile and Dimitri Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations,, \emph{Adv. Nonlinear Stud.}, 4 (2004), 307. Google Scholar [23] Teresa D'Aprile and Dimitri Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 134 (2004), 893. doi: 10.1017/S030821050000353X. Google Scholar [24] Teresa D'Aprile and Juncheng Wei, Layered solutions for a semilinear elliptic system in a ball,, \emph{J. Differential Equations}, 226 (2006), 269. doi: 10.1016/j.jde.2005.12.009. Google Scholar [25] Teresa D'Aprile and Juncheng Wei, Solutions en grappe autour des centres harmoniques d'un système elliptique couplé,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 24 (2007), 605. doi: 10.1016/j.anihpc.2006.04.003. Google Scholar [26] Pietro d'Avenia and Lorenzo Pisani, Nonlinear Klein-Gordon equations coupled with Born-Infeld type equations,, \emph{Electron. J. Differential Equations}, (2002). Google Scholar [27] P. d'Avenia, L. Pisani and G. Siciliano, Dirichlet and Neumann problems for Klein-Gordon-Maxwell systems,, \emph{Nonlinear Anal.}, 71 (2009). doi: 10.1016/j.na.2009.02.111. Google Scholar [28] Pietro d'Avenia, Lorenzo Pisani and Gaetano Siciliano, Klein-Gordon-Maxwell systems in a bounded domain,, \emph{Discrete Contin. Dyn. Syst.}, 26 (2010), 135. doi: 10.3934/dcds.2010.26.135. Google Scholar [29] Pierpaolo Esposito, Angela Pistoia and Jérôme Vétois, The effect of linear perturbations on the Yamabe problem,, \emph{Math. Ann.}, (). doi: 10.1007/s00208-013-0971-9. Google Scholar [30] Vladimir Georgiev and Nicola Visciglia, Solitary waves for Klein-Gordon-Maxwell system with external Coulomb potential,, \emph{J. Math. Pures Appl.}, 84 (2005), 957. doi: 10.1016/j.matpur.2004.09.016. Google Scholar [31] Marco Ghimenti and Anna Maria Micheletti, Number and profile of low energy solutions for singularly perturbed Klein-Gordon-Maxwell systems on a Riemannian manifold,, Preprint, (2013). doi: 10.1016/j.jde.2014.01.012. Google Scholar [32] B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations,, \emph{Comm. Partial Differential Equations}, 6 (1981), 883. doi: 10.1080/03605308108820196. Google Scholar [33] David Gilbarg and Neil S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Classics in Mathematics, (2001). Google Scholar [34] Emmanuel Hebey, Solitary waves in critical abelian gauge theories,, \emph{Discrete Contin. Dyn. Syst.}, 32 (2012), 1747. doi: 10.3934/dcds.2012.32.1747. Google Scholar [35] Emmanuel Hebey, Compactness and Stability for Nonlinear Elliptic Equations,, European Mathematical Society, (). doi: 10.4171/134. Google Scholar [36] Emmanuel Hebey and Juncheng Wei, Resonant states for the static Klein-Gordon-Maxwell-Proca system,, \emph{Math. Res. Lett.}, 19 (2012), 953. doi: 10.4310/MRL.2012.v19.n4.a18. Google Scholar [37] Emmanuel Hebey and Juncheng Wei, Schrödinger-Poisson systems in the 3-sphere,, \emph{Calc. Var. Partial Differential Equations}, 47 (2013), 25. doi: 10.1007/s00526-012-0509-0. Google Scholar [38] Emmanuel Hebey and Trong Tuong Truong, Static Klein-Gordon-Maxwell-Proca systems in 4-dimensional closed manifolds,, \emph{J. Reine Angew. Math., 667 (2012), 221. Google Scholar [39] Emmanuel Hebey and Michel Vaugon, The best constant problem in the Sobolev embedding theorem for complete Riemannian manifolds,, \emph{Duke Math. J.}, 79 (1995), 235. doi: 10.1215/S0012-7094-95-07906-X. Google Scholar [40] YanYan Li and Lei Zhang, Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations,, \emph{J. Anal. Math.}, 90 (2003), 27. doi: 10.1007/BF02786551. Google Scholar [41] YanYan Li and Lei Zhang, A Harnack type inequality for the Yamabe equation in low dimensions,, \emph{Calc. Var. Partial Differential Equations}, 20 (2004), 133. doi: 10.1007/s00526-003-0224-y. Google Scholar [42] YanYan Li and Lei Zhang, Compactness of solutions to the Yamabe problem. II,, \emph{Calc. Var. Partial Differential Equations}, 24 (2005), 185. doi: 10.1007/s00526-004-0320-7. Google Scholar [43] Yanyan Li and Meijun Zhu, Yamabe type equations on three-dimensional Riemannian manifolds,, \emph{Commun. Contemp. Math.}, 1 (1999), 1. doi: 10.1142/S021919979900002X. Google Scholar [44] Dimitri Mugnai, Coupled Klein-Gordon and Born-Infeld-type equations: looking for solitary waves,, \emph{Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.}, 460 (2004), 1519. doi: 10.1098/rspa.2003.1267. Google Scholar [45] Dimitri Mugnai, Solitary waves in abelian gauge theories with strongly nonlinear potentials,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 27 (2010), 1055. doi: 10.1016/j.anihpc.2010.02.001. Google Scholar [46] Frédérique Robert and Jérôme Vétois, Examples of non-isolated blow-up for perturbations of the scalar curvature equation,, Preprint, (2012). Google Scholar [47] Frédérique Robert and Jérôme Vétois, A General Theorem for the Construction of Blowing-up Solutions to Some Elliptic Nonlinear Equations with Lyapunov-Schmidt's Finite-dimensional Reduction, Cocompact Imbeddings, Profile Decompositions, and their Applications to PDE,, Trends Math., (2013), 85. Google Scholar [48] Michael Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities,, \emph{Math. Z.}, 187 (1984), 511. doi: 10.1007/BF01174186. Google Scholar [49] Neil S. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds,, \emph{Ann. Scuola Norm. Sup. Pisa}, 22 (1968), 265. Google Scholar
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