On nodal radial solutions of an elliptic problem involving critical Sobolev exponent

summary:In this paper we construct radial solutions of equation (1) (and (13)) having prescribed number of nodes

Magnetic Monopoles, Electric Neutrality and the Static Maxwell-Dirac Equations

We study the full Maxwell-Dirac equations: Dirac field with minimally coupled electromagnetic field and Maxwell field with Dirac current as source. Our particular interest is the static case in which the Dirac current is purely time-like -- the "electron" is at rest in some Lorentz frame. In this case we prove two theorems under rather general assumptions. Firstly, that if the system is also stationary (time independent in some gauge) then the system as a whole must have vanishing total charge, i.e. it must be electrically neutral. In fact, the theorem only requires that the system be {\em asymptotically} stationary and static. Secondly, we show, in the axially symmetric case, that if there are external Coulomb fields then these must necessarily be magnetically charged -- all Coulomb external sources are electrically charged magnetic monopoles

Existence results for quasilinear Dirichlet problem

This paper deals with the Dirichlet problem (1) -Σ D(a(x, u) D1u) + c(x)u = b(x, u, Du) in Q. (2) n(x)=ф(x) on ∂Q. in a bounded domain Q⊂R,n with the boundary ∂Q of class C2 and a function ф which, in general, ..

On symmetric solutions of a critical semilinear elliptic system involving the Caffarelli-Kohn-Nirenberg inequality in R N RN\mathbb{R}^{N}

Abstract This paper deals with the existence and multiplicity of symmetric solutions for a weighted semilinear elliptic system with multiple critical Hardy-Sobolev exponents and singular potentials in R N $\mathbb{R}^{N}$ . Applying the symmetric criticality principle and Caffarelli-Kohn-Nirenberg inequality, we establish several existence and multiplicity results of G-symmetric solutions under certain appropriate hypotheses on the parameters and the weighted functions