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Register a new userMulti-bump solutions of $-Delta u=K(x)u^{frac{n+2}{n-2}}$ on lattices in $R^n$
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We consider critical exponent semi-linear elliptic equation with coefficient K(x) periodic in its first k variables, with 2k smaller than n-2. Under some natural conditions on K near a critical point, we prove the existence of multi-bump solutions where the centers of bumps can be placed in some lattices in Rk, including infinite lattices. We also show that for 2k greater than or equal to n-2, no such solutions exist.
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This note shows that under $(p,alpha, N)in (1,infty)times(0,2)timesmathbb Z_+$ the fractional order differential inequality $$ (dagger)quad u^p le (-Delta)^{frac{alpha}{2}} uquadhbox{in}quadmathbb R^{N} $$ has the property that if $Nlealpha$ then a nonnegative solution to $(dagger)$ is unique, and if $N>alpha$ then the uniqueness of a nonnegative weak solution to $(dagger)$ occurs when and only when $ple N/(N-alpha)$, thereby innovatively generalizing Gidas-Sprucks result for $u^p+Delta ule 0$ in $R^N$ discovered in cite{GS}.
We study N = 2* theories with gauge group U(N) and use equivariant localization to calculate the quantum expectation values of the simplest chiral ring elements. These are expressed as an expansion in the mass of the adjoint hypermultiplet, with coefficients given by quasi-modular forms of the S-duality group. Under the action of this group, we construct combinations of chiral ring elements that transform as modular forms of definite weight. As an independent check, we confirm these results by comparing the spectral curves of the associated Hitchin system and the elliptic Calogero-Moser system. We also propose an exact and compact expression for the 1-instanton contribution to the expectation value of the chiral ring elements.
Strings in $mathcal{N}=2$ supersymmetric ${rm U}(1)^N$ gauge theories with $N$ hypermultiplets are studied in the generic setting of an arbitrary Fayet-Iliopoulos triplet of parameters for each gauge group and an invertible charge matrix. Although the string tension is generically of a square-root form, it turns out that all existing BPS (Bogomolnyi-Prasad-Sommerfield) solutions have a tension which is linear in the magnetic fluxes, which in turn are linearly related to the winding numbers. The main result is a series of theorems establishing three different kinds of solutions of the so-called constraint equations, which can be pictured as orthogonal directions to the magnetic flux in ${rm SU}(2)_R$ space. We further prove for all cases, that a seemingly vanishing Bogomolnyi bound cannot have solutions. Finally, we write down the most general vortex equations in both master form and Taubes-like form. Remarkably, the final vortex equations essentially look Abelian in the sense that there is no trace of the ${rm SU}(2)_R$ symmetry in the equations, after the constraint equations have been solved.
In this paper we construct N=(1,0) and N=(1,1/2) non-singlet Q-deformed supersymmetric U(1) actions in components. We obtain an exact expression for the enhanced supersymmetry action by turning off particular degrees of freedom of the deformation tensor. We analyze the behavior of the action upon restoring weekly some of the deformation parameters, obtaining a non trivial interaction term between a scalar and the gauge field, breaking the supersymmetry down to N=(1,0). Additionally, we present the corresponding set of unbroken supersymmetry transformations. We work in harmonic superspace in four Euclidean dimensions.
It is shown that the mass dependence of the $Lambda$-lifetime in heavy hypernuclei is sensitive to the ratio of neutron-induced to proton-induced non-mesonic decay rates R_n/R_p. A comparison of the experimental mass dependence of the lifetimes with the calculated ones for different values of R_n/R_p leads to the conclusion that this ratio is larger than 2 on the confidence level of 0.75. This suggests that the phenomenological $Delta$I=1/2 rule might be violated for the nonmesonic decay of the $Lambda$-hyperon.
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