Results 1  10
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18
Singular limits in Liouvilletype equations
 Calc. Var. Partial Differential Equations
"... Abstract. We consider the boundary value problem ∆u+ε2 k(x) eu = 0 in a bounded, smooth domain Ω in R2 with homogeneous Dirichlet boundary conditions. Here ε> 0, k(x) is a nonnegative, not identically zero function. We find conditions under which there exists a solution uε which blows up at exac ..."
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Cited by 14 (4 self)
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Abstract. We consider the boundary value problem ∆u+ε2 k(x) eu = 0 in a bounded, smooth domain Ω in R2 with homogeneous Dirichlet boundary conditions. Here ε> 0, k(x) is a nonnegative, not identically zero function. We find conditions under which there exists a solution uε which blows up at exactly m points as ε → 0 and satisfies ε2 ∫ Ω keuε → 8mpi. In particular, we find that if k ∈ C2(Ω̄), infΩ k> 0 and Ω is not simply connected then such a solution exists for any given m ≥ 1
Asymptotic Behavior of Blowup Solutions for Elliptic Equations with Exponential Nonlinearity and Singular Data
, 810
"... We consider a sequence of blowup solutions of a two dimensional, second order elliptic equation with exponential nonlinearity and singular data. This equation has a rich background in physics and geometry. In a work of BartolucciChenLinTarantello it is proved that the profile of the solutions dif ..."
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Cited by 2 (2 self)
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We consider a sequence of blowup solutions of a two dimensional, second order elliptic equation with exponential nonlinearity and singular data. This equation has a rich background in physics and geometry. In a work of BartolucciChenLinTarantello it is proved that the profile of the solutions differs from global solutions of a Liouville type equation only by a uniformly bounded term. The present paper improves their result and establishes an expansion of the solutions near the blowup points with a sharp error estimate.
A system of elliptic equations arising in Chern–Simons field theory
 J. Funct. Anal
"... Abstract. We prove the existence of topological vortices in a relativistic selfdual Abelian ChernSimons theory with two Higgs particles and two gauge fields through a study of a coupled system of two nonlinear elliptic equations over R2. We present two approaches to prove existence of solutions on ..."
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Cited by 2 (1 self)
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Abstract. We prove the existence of topological vortices in a relativistic selfdual Abelian ChernSimons theory with two Higgs particles and two gauge fields through a study of a coupled system of two nonlinear elliptic equations over R2. We present two approaches to prove existence of solutions on bounded domains: via minimization of an indefinite functional and via a fixed point argument. We then show that we may pass to the full R2 limit from the boundeddomain solutions to obtain a topological solution in R2. Contents
Existence of multistring solutions of the selfgravitating massive
 W boson, Lett. Math. Phys
"... We consider a semilinear elliptic system which include the model system of the W −strings in the cosmology as a special case. We prove existence of multistring solutions and obtain precise asymptotic decay estimates near infinity for the solutions. As a special case of this result we solve an open ..."
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We consider a semilinear elliptic system which include the model system of the W −strings in the cosmology as a special case. We prove existence of multistring solutions and obtain precise asymptotic decay estimates near infinity for the solutions. As a special case of this result we solve an open problem posed in [13] Key Words: semilinear elliptic system, exponential nonlinearities, selfdual gauge field theories
Selfgravitating Electroweak strings
"... We obtain selfgravitating multistring configurations for the EinsteinWeinbergSalam model, in terms of solutions for a nonlinear elliptic system of Liouville type whose solvability was posed as an open problem in [15]. 1 ..."
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We obtain selfgravitating multistring configurations for the EinsteinWeinbergSalam model, in terms of solutions for a nonlinear elliptic system of Liouville type whose solvability was posed as an open problem in [15]. 1
HalfSkyrmions and SpikeVortex Solutions of TwoComponent Nonlinear Schrödinger Systems
, 2007
"... Recently, skyrmions with integer topological charges have been observed numerically but have not yet been shown rigorously on twocomponent systems of nonlinear Schrödinger equations (NLSE) describing a binary mixture of BoseEinstein condensates (cf. [2] and [25]). Besides, halfskyrmions character ..."
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Recently, skyrmions with integer topological charges have been observed numerically but have not yet been shown rigorously on twocomponent systems of nonlinear Schrödinger equations (NLSE) describing a binary mixture of BoseEinstein condensates (cf. [2] and [25]). Besides, halfskyrmions characterized by halfinteger topological charges can also be found in the nonlinear σ model which is a model of the BoseEinstein condensate of the Schwinger bosons (cf. [18]). Here we prove rigorously the existence of halfskyrmions which may come from a new type of soliton solutions called spikevortex solutions of twocomponent systems of NLSE on the entire plane R 2. These spikevortex solutions having spikes in one component and a vortex in the other component may form halfskyrmions. By LiapunovSchmidt reduction process, we may find spikevortex solutions of twocomponent systems of NLSE. 1
On a Liouvilletype equation with signchanging weight
, 2007
"... In this paper we study the existence, nonexistence and multiplicity of nonnegative solutions for the family of problems −∆u = λ (a(x)e u + f(x, u)), u ∈ H 1 0 (Ω) where Ω is a bounded domain in R 2 and λ> 0 is a parameter. The coefficient a(x) is allowed to change sign. The techniques used in th ..."
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In this paper we study the existence, nonexistence and multiplicity of nonnegative solutions for the family of problems −∆u = λ (a(x)e u + f(x, u)), u ∈ H 1 0 (Ω) where Ω is a bounded domain in R 2 and λ> 0 is a parameter. The coefficient a(x) is allowed to change sign. The techniques used in the proofs are a combination of upper and lower solutions, the TrudingerMoser inequality and variational methods. Note that when f(x, u) = 0 the equation is of Liouville type.
Concentrating solutions for the Hénon equation 2 ∗ in IR
, 2005
"... We consider the boundary value problem ∆u + x  2α u p = 0, α> 0, in the unit ball B with homogeneous Dirichlet boundary condition and p a large exponent. We find a condition which ensures the existence of a positive solution up concentrating outside the origin at k symmetric points as p goes to ..."
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We consider the boundary value problem ∆u + x  2α u p = 0, α> 0, in the unit ball B with homogeneous Dirichlet boundary condition and p a large exponent. We find a condition which ensures the existence of a positive solution up concentrating outside the origin at k symmetric points as p goes to +∞. The same techniques lead also to a more general result on general domains. In particular, we have that concentration at the origin is always possible, provided α / ∈ IN.
On the BornInfeld Abelian Higgs cosmic strings with symmetric vacuum
"... We study a system of elliptic equations from the Abelian BornInfeld system coupled with the Einstein equations under the boundary condition of the symmetric vacuum(nontopological type). When the total string number satisfies 1 ≤ N < 1 4πG, where G is the gravitational constant, we construct a fa ..."
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We study a system of elliptic equations from the Abelian BornInfeld system coupled with the Einstein equations under the boundary condition of the symmetric vacuum(nontopological type). When the total string number satisfies 1 ≤ N < 1 4πG, where G is the gravitational constant, we construct a family of solutions to the system. The qualitative properties of the solutions are quite different from the solutions with the boundary condition of the broken vacuum symmetry.