DCDS
Solitary waves in critical Abelian gauge theories
Emmanuel Hebey
Discrete & Continuous Dynamical Systems - A 2012, 32(5): 1747-1761 doi: 10.3934/dcds.2012.32.1747
We prove existence of standing waves solutions for electrostatic Klein-Gordon-Maxwell systems in arbitrary dimensional compact Riemannian manifolds with boundary for zero Dirichlet boundary conditions. We prove that phase compensation holds true when the dimension $n = 3$ or $4$. In these dimensions, existence of a solution is obtained when the mass of the particle field, balanced by the phase, is small in a geometrically quantified sense. In particular, existence holds true for sufficiently large phases. When $n \ge 5$, existence of a solution is obtained when the mass of the particle field is sufficiently small.
keywords: phase compensation. Solitary waves Klein-Gordon-Maxwell systems
CPAA
Multiple solutions for critical elliptic systems in potential form
Emmanuel Hebey Jérôme Vétois
Communications on Pure & Applied Analysis 2008, 7(3): 715-741 doi: 10.3934/cpaa.2008.7.715
We discuss and prove existence of multiple solutions for critical elliptic systems in potential form on compact Riemannian manifolds.
keywords: Changing sign solutions Lusternik-Schnirelmann category. critical systems
CPAA
The Lin-Ni's conjecture for vector-valued Schrödinger equations in the closed case
Emmanuel Hebey
Communications on Pure & Applied Analysis 2010, 9(4): 955-962 doi: 10.3934/cpaa.2010.9.955
We prove that critical vector-valued Schrödinger equations on compact Riemannian manifolds possess only constant solutions when the potential is sufficiently small. We prove the result in dimension $n = 3$ for arbitrary manifolds and in dimension $n \ge 4$ for manifolds with positive curvature. We also establish a gap estimate on the smallness of the potentials for the specific case of $S^1(T)\times S^{n-1}$.
keywords: Critical equations Riemannian manifolds Lin-Ni's conjecture vector-valued equations

Year of publication

Related Authors

Related Keywords

[Back to Top]