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In this paper, we consider the self-dual O (3) Maxwell-Chern-Simons-Higgs equations on a two-dimensional flat torus arising from the O (3) sigma gauge field model. Our main goal is to obtain the existence and uniqueness of topological solutions with vortices and antivortices. In order to achieve this goal, we show that the nondegeneracy of linearized operator for entire solution holds even when the symmetry is broken. Moreover, we also obtain the uniform bound of L 1 norm of nonlinearities with respect to large charge of electron. We expect that these results would play an significant role to get the asymptotic behavior of all possible solutions and count the total number of solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the degree counting formula of the rank two Toda system with simple singular source when ρ 1 ∈ (0 , 4 π) ∪ (4 π , 8 π) and ρ 2 ∉ 4 π N. The key step is to derive the degree formula of the shadow system, which arises from the bubbling solutions as ρ 1 tends to 4 π. In order to compute the topological degree of the shadow system, we need to find some suitable deformation. During this deformation, we shall deal with new difficulty arising from the phenomenon: blow up does not necessarily imply concentration of mass. This phenomenon occurs due to the collapsing of singularities. This is a continuation of the previous work [25]. [ABSTRACT FROM AUTHOR]

The seminal work [7] by Brezis and Merle showed that the bubbling solutions of the mean field equation have the property of mass concentration. Recently, Lin and Tarantello in [30] found that the "bubbling implies mass concentration" phenomena might not hold if there is a collapse of singularities. Furthermore, a sharp estimate [23] for the bubbling solutions has been obtained. In this paper, we prove that there exists at most one sequence of bubbling solutions if the collapsing singularity occurs. The main difficulty comes from that after re-scaling, the difference of two solutions locally converges to an element in the kernel space of the linearized operator. It is well-known that the kernel space is three dimensional. So the main technical ingredient of the proof is to show that the limit after re-scaling is orthogonal to the kernel space. [ABSTRACT FROM AUTHOR]

We initiate the program for computing the Leray–Schauder topological degree for Toda systems of rank two. This program still contains a lot of challenging problems for analysts. As the first step, we prove that if a sequence of solutions ( u 1 k , u 2 k ) blows up, then one of h j e u j k ∫ M h j e u j k d v g , j = 1 , 2 tends to a sum of Dirac measures. This is so-called the phenomena of weak concentration. Our purposes in this article are (i) to introduce the shadow system due to the bubbling phenomena when one of parameters ρ i crosses 4 π and ρ j ∉ 4 π N where 1 ≤ i ≠ j ≤ 2 ; (ii) to show how to calculate the topological degree of Toda systems by computing the topological degree of the general shadow systems; (iii) to calculate the topological degree of the shadow system for one point blow up. We believe that the degree counting formula for the shadow system would be useful in other problems. [ABSTRACT FROM AUTHOR]

In this paper, we consider B 2 and G 2 Toda systems on a compact Riemann surface M. We investigate the relation between the topological property of M and the Leray-Schauder degree of the Toda systems by computing the degree jump caused by multi-bubbling phenomena of the Toda system with non-symmetric Cartan matrix. The bubbling phenomena can reduce the computations of this degree jump to the calculations of the topological degree for some mean field equation together with an additional condition, which is called the shadow system in this paper. The main purpose of this paper is to compute the topological degree of this shadow system via a suitable deformation from a "decoupled" shadow system to our shadow system, and to show the a priori bound along this deformation. The proof of a priori bounds is very subtle, which depends on the type of Lie algebra. [ABSTRACT FROM AUTHOR]

Abstract: Let Ω be a bounded domain in with the boundary . We consider the following singularly perturbed nonlinear elliptic problem on Ω, where ν is the exterior normal to ∂Ω and the nonlinearity f is of subcritical growth. It has been known that under Berestycki and Lions conditions for and , there exists a solution of the problem which develops a spike layer near a local maximum point of the mean curvature H on ∂Ω for small . In this paper, we extend the previous result for and . [Copyright &y& Elsevier]

Abstract: Eicosapentaenoic acid (EPA) is an omega-3 (ω-3) polyunsaturated fatty acid (PUFA), which has anti-inflammatory and anti-cancer properties. Some reports have demonstrated that EPA inhibits NF-κB activation induced by tumor necrosis factor (TNF)-α or lipopolysaccharide (LPS) in various cells. However, its detailed mode of action is unclear. In this report, we investigated whether EPA inhibits the expression of TNF-α-induced matrix metalloproteinases (MMP)-9 in human immortalized keratinocytes (HaCaT). TNF-α induced MMP-9 expression by NF-κB-dependent pathway. Pretreatment of EPA inhibited TNF-α-induced MMP-9 expression and p65 phosphorylation. However, EPA could not affect IκB-α phosphorylation, nuclear translocation of p65, and DNA binding activity of NF-κB. EPA inhibited TNF-α-induced p65 phosphorylation through p38 and Akt inhibition and this inhibition was IKKα-dependent event. Taken together, we demonstrate that EPA inhibits TNF-α-induced MMP-9 expression through inhibition of p38 and Akt activation. [Copyright &y& Elsevier]

In this paper, we prove a partly clustering phenomenon for nonlinear Schrödinger systems with large mixed couplings of attractive and repulsive forces, which arise from the models in Bose-Einstein condensates and nonlinear optics. More precisely, we consider a system with three components where the interaction between the first two components and the third component is repulsive, and the interaction between the first two components is attractive. Recent studies [10–13] in this case show that for large interaction forces, the first two components are localized in a region with a small energy and the third component is close to a solution of a single equation. Especially, the results in the works [12,13] say that the region of localization for a (locally) least energy vector solution on a ball in the class of radially symmetric functions is the origin or the whole boundary depending on the space dimension 1 ≤ n ≤ 3. In this paper we construct a new type of solutions with a region of localization different from the origin or the whole boundary. In fact, we show that there exist radially symmetric positive vector solutions with clustering multi-bumps for the first two components near the maximum point of r n − 1 U 3 , where U is the limit of the third component and the maximum point is the only critical point different from the origin and the boundary. [ABSTRACT FROM AUTHOR]

We consider non-topological solutions of a nonlinear elliptic system problem (see (1.4) below) derived from the S U (3) Chern-Simons models in R 2. The existence of non-topological solutions even for radial symmetric case has been a long standing open problem. Recently, Choe, Kim, and Lin in [7,8] showed the existence of radial symmetric non-topological solution when the vortex points collapse. However, the arguments in [7,8] cannot work for an arbitrary configuration of vortex points. In this paper, we develop a new approach by using different scalings for different components of the system to construct a family of non-topological solutions, which blows up at infinity. [ABSTRACT FROM AUTHOR]

We are concerned with the mean field equation with singular data on bounded domains. By assuming a singular point to be a critical point of the 1-vortex Kirchhoff-Routh function, we prove local uniqueness and non-degeneracy of bubbling solutions blowing up at a singular point. The proof is based on sharp estimates for bubbling solutions of singular mean field equations and a suitably defined Pohozaev-type identity. [ABSTRACT FROM AUTHOR]

We consider the self-dual equations arising from the Chern–Simons gauge theory of rank 2 such as the S U ( 3 ) , S O ( 5 ) , and G 2 Chern–Simons model in R 2 . There are three possible types of solutions in these theories, namely, topological, nontopological, and mixed type solutions. The existence of a mixed type solution for an arbitrary configuration of vortex points has been a long-standing open problem. We construct here a family of mixed type solutions ( u 1 , ε , u 2 , ε ) , ε > 0 for an arbitrary configuration of vortex points under a mild non-degeneracy condition. The main new idea of our construction of an approximate solution is to use different scalings for different components. In the process of the finite dimensional reduction, all estimates for this particular construction become rather delicate. [ABSTRACT FROM AUTHOR]