2009, 23(3): 655-683. doi: 10.3934/dcds.2009.23.655

On the prescribed scalar curvature on $3$-half spheres: Multiplicity results and Morse inequalities at infinity

1. 

Département de Mathématiques, Faculté des Sciences de Sfax, Route Soukra, Sfax, Tunisia

2. 

Mathematisches institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tubingen, Germany

Received  August 2007 Revised  July 2008 Published  November 2008

We consider the existence and multiplicity of riemannian metrics of prescribed mean curvature and zero boundary mean curvature on the three dimensional half sphere $(S^3_+,g_c)$ endowed with its standard metric $g_c$. Due to Kazdan-Warner type obstructions, conditions on the function to be realized as a scalar curvature have to be given. Moreover the existence of critical point at infinity for the associated Euler Lagrange functional makes the existence results harder to be proved. However it turns out that such noncompact orbits of the gradient can be treated as a usual critical point once a Morse Lemma at infinity is performed. In particular their topological contribution to the level sets of the functional can be computed. In this paper we prove that, under generic conditions on $K$, this topology at infinity is a lower bound for the number of metrics in the conformal class of $g_c$ having prescribed scalar curvature and zero boundary mean curvature.
Citation: M. Ben Ayed, Mohameden Ould Ahmedou. On the prescribed scalar curvature on $3$-half spheres: Multiplicity results and Morse inequalities at infinity. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 655-683. doi: 10.3934/dcds.2009.23.655
[1]

Yoshikazu Giga, Yukihiro Seki, Noriaki Umeda. On decay rate of quenching profile at space infinity for axisymmetric mean curvature flow. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1463-1470. doi: 10.3934/dcds.2011.29.1463

[2]

Mohameden Ahmedou, Mohamed Ben Ayed, Marcello Lucia. On a resonant mean field type equation: A "critical point at Infinity" approach. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1789-1818. doi: 10.3934/dcds.2017075

[3]

Zhirong He, Weinian Zhang. Critical periods of a periodic annulus linking to equilibria at infinity in a cubic system. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 841-854. doi: 10.3934/dcds.2009.24.841

[4]

Luis Barreira, Claudia Valls. Topological conjugacies and behavior at infinity. Communications on Pure & Applied Analysis, 2014, 13 (2) : 687-701. doi: 10.3934/cpaa.2014.13.687

[5]

Jaume Llibre, Jesús S. Pérez del Río, J. Angel Rodríguez. Structural stability of planar semi-homogeneous polynomial vector fields applications to critical points and to infinity. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 809-828. doi: 10.3934/dcds.2000.6.809

[6]

Yinbin Deng, Yi Li, Wei Shuai. Existence of solutions for a class of p-Laplacian type equation with critical growth and potential vanishing at infinity. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 683-699. doi: 10.3934/dcds.2016.36.683

[7]

Mingwen Fei, Huicheng Yin. Nodal solutions of 2-D critical nonlinear Schrödinger equations with potentials vanishing at infinity. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2921-2948. doi: 10.3934/dcds.2015.35.2921

[8]

Yinbin Deng, Wei Shuai. Positive solutions for quasilinear Schrödinger equations with critical growth and potential vanishing at infinity. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2273-2287. doi: 10.3934/cpaa.2014.13.2273

[9]

Jingxian Sun, Shouchuan Hu. Flow-invariant sets and critical point theory. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 483-496. doi: 10.3934/dcds.2003.9.483

[10]

Victor S. Kozyakin, Alexander M. Krasnosel’skii, Dmitrii I. Rachinskii. Arnold tongues for bifurcation from infinity. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 107-116. doi: 10.3934/dcdss.2008.1.107

[11]

Victor Kozyakin, Alexander M. Krasnosel’skii, Dmitrii Rachinskii. Asymptotics of the Arnold tongues in problems at infinity. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 989-1011. doi: 10.3934/dcds.2008.20.989

[12]

Michel Chipot, Aleksandar Mojsic, Prosenjit Roy. On some variational problems set on domains tending to infinity. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3603-3621. doi: 10.3934/dcds.2016.36.3603

[13]

Guillaume James, Dmitry Pelinovsky. Breather continuation from infinity in nonlinear oscillator chains. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1775-1799. doi: 10.3934/dcds.2012.32.1775

[14]

Yukihiro Seki. A remark on blow-up at space infinity. Conference Publications, 2009, 2009 (Special) : 691-696. doi: 10.3934/proc.2009.2009.691

[15]

Francesco Della Pietra, Ireneo Peral. Breaking of resonance for elliptic problems with strong degeneration at infinity. Communications on Pure & Applied Analysis, 2011, 10 (2) : 593-612. doi: 10.3934/cpaa.2011.10.593

[16]

Begoña Alarcón, Víctor Guíñez, Carlos Gutierrez. Hopf bifurcation at infinity for planar vector fields. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 247-258. doi: 10.3934/dcds.2007.17.247

[17]

Alexander Krasnosel'skii, Jean Mawhin. The index at infinity for some vector fields with oscillating nonlinearities. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 165-174. doi: 10.3934/dcds.2000.6.165

[18]

Yaiza Canzani, Dmitry Jakobson, Igor Wigman. Scalar curvature and $Q$-curvature of random metrics. Electronic Research Announcements, 2010, 17: 43-56. doi: 10.3934/era.2010.17.43

[19]

Matteo Novaga, Enrico Valdinoci. Closed curves of prescribed curvature and a pinning effect. Networks & Heterogeneous Media, 2011, 6 (1) : 77-88. doi: 10.3934/nhm.2011.6.77

[20]

Yong Huang, Lu Xu. Two problems related to prescribed curvature measures. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1975-1986. doi: 10.3934/dcds.2013.33.1975

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (2)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]