On the family of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory

In this paper, by constructing a family of approximation solutions and applying a specific version of the Implicit Function Theorem (please see, e.g. [ 18 ]), we prove the existence of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory.

Here we use the Einstein summation convention, the Greek letters, µ and ν run from 0 to 3 while the roman letters j and k are from 1 to 3. Following [19], we consider two Abelian gauge fields,Â µ andÃ µ , generated from the product gauge group U (1) ×Ũ (1), and use q and p to denote two charged scalars carrying charge (+1, 1) and (0, +1), respectively, so that the gauge-covariant derivatives are given by Using the spacetime metric tensor g µν of signature (+ − −−) to raise and lower indices, we rewrite the Lagrangian density of the TongWong model [19] as where ζ,ζ > 0 are parameters,F µν ,F µν are electromagnetic fields induced from gauge potentialsÂ µ ,Ã µ . Taking the variation of the Einstein-Hilbert action where R g is the Ricci scalar curvature of the metric g µν , G > 0 the universal gravitational constant, and Λ the cosmological constant, the equations of the motion of the action (4) are where G µν is the Einstein tensor and T µν the energy-momentum tensor of the matter and gauge field sector given by As in [13,17] we look for straight time independent cosmic string solutions so that the spacetime is uniform along the time axis x 0 = t and the x 3 -direction and the line element takes the form where now g jk is the Riemannian metric tensor of an orientable 2-surface M with local coordinates x 1 , x 2 . Within such a metric the only nontrivial components of the Einstein tensor are G 00 = −G 33 = −K g , where K g is the Gauss curvature of the surface (M, {g jk }), which imposes constraints to the form of the energy-momentum tensor via (5). On the other hand, it is natural and compatible to assume that the gauge and scalar fields depend on the coordinates on M only and that the 0-and 3-components of the gauge fields are zero. Thus, we have T 03 = T 0j = T 3j = 0 and T 00 = −T 33 = H immediately, where H is the Hamiltonian of the Tong-Wong model (3) given by To facilitate our computation, recall that the associated current densities satisfy the identities Applying (13) and (14), we may rewrite (11) as where ∇ j is the covariant derivative with respect to the metric g jk over M and jk is the Kronecker skew-symmetric tensor with 12 = |g| in which |g| = det(g jk ).
In view of [19], we designateÛ (1) andŨ (1) to be the magnetic fluxes given by wherek,k are integers. we obtain from integrating (15) the energy lower bound The lower bound in (16) is attained when the equations are satisfied for j, k = 1, 2 withk = ±|k|,k = ±|k|. As a consequence of (17), it is direct to check that [15,19,20] T jk = 0 (j, k = 1, 2). Inserting this result into (5) we arrive at the vanishing cosmological constant condition Λ = 0. Therefore, Equation (5) becomes In view of (11)and (17), we have where ∆ g is the Laplace-Beltrami operator induced from covariant derivative ∇ j such that ∆ g f = ∇ j ∇ j f = (1/ |g|)∂ j (g jk |g|∂ k f ). Let n,ñ ≥ 0 denote the winding numbers of q and p, respectively. Then n,ñ are the algebraic numbers of zeros of q, p, respectively, which are related to the magnetic flux numbersk,k by n =k −k,ñ =k as indicated in [19]. The sets of zeros of q, p are denoted as Set u = log |q| 2 and v = log |p| 2 . It is standard that (17) may be recast into where δ z denotes the Dirac function defined over the 2-surface (M, {g jk }) and concentrated at the point z ∈ M.
Let g jk be a unknown metric which is conformal to a known one, g 0,jk , so that g jk = e η g 0,jk (j, k = 1, 2). Then we have the relations In view of the second relation in (21), we see that the system (20) becomes where the Dirac functions are defined over the 2-surface (M, {g 0,jk }) instead. From (18), (19) and (21), we have We consider the situation when M is non-compact. If M is compact, we refer the readers to the results of [8] and the references therein. For simplicity, we assume that M = R 2 and g jk = e η δ jk . Now g 0,jk = δ jk and ∆ g0 = ∆ is the usual Laplace operator on R 2 . Equation (22) becomes where the Dirac δ-functions are defined over R 2 . Note that, in (23), we have K g0 = 0. Hence we obtain In order to facilitate the description of the asymptotic behavior of the fields at infinity, one may take the substitution So (24) is updated into Correspondingly, (25) is modified into From (27) and (28), it is not hard to see that [20] the gravitational conformal factor e η is exactly determined by the expression (2).
After Theorem 1.1, there is an intersecting problem about finding the sharp range of flux for all solutions of (1) satisfy one of (29), (30) and (31). Recently, there are many well-known results about this problem for another single and coupled equations in [1,9,10,11,12,16].
The present paper is organized as follows. In the following section we construct a family of the non-topological solutions of (32) and carry out the main theorem from an approximation viewpoint.

ON THE FAMILY OF NON-TOPOLOGICAL SOLUTIONS 3299
The last two formulas together with (44) give In view of (48), (49) and (50), we introduce the following mappings: Now we let the functions w 1 , w 2 , w 3 be given as follows: Function Spaces. Define the scalar products ·, · Xα and ·, · Yα , 0 < α < 1/2, for functions in the spaces L 2 loc (R 2 ) and W 2,2 loc (R 2 ): Let X α , Y α be the completion of L 2 loc (R 2 ) and W 2,2 loc (R 2 ) by the norm · Xα and · Yα , respectively, where α . Note that X α → L 1 (R 2 ) and Y α ⊂ C 0 loc (R 2 ) from Hölder's inequality as well as the local regularity of the Laplace operator. Moreover, there exists C > 0 such that for all u ∈ Y α , where log + |x| = max{ 0, log |x|} (please see [2]). We denote the subspaces consisting of radially symmetric functions by X r α and Y r α .