CPAA
Multiple solutions for critical elliptic systems in potential form
Emmanuel Hebey Jérôme Vétois
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
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
DCDS
Solitary waves in critical Abelian gauge theories
Emmanuel Hebey
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

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