Multiplicity for a nonlinear elliptic fourth order equation in Maxwell-Chern-Simons vortex theory

We prove the existence of at least two solutions for a fourth order equation, which includes the vortex equations for the U(1) and CP(1) self-dual Maxwell-Chern-Simons models as special cases. Our method is variational, and it relies on an "asymptotic maximum principle" property for a special class of supersolutions to this fourth order equation.Comment: 20 page

A sharp Sobolev inequality on Riemannian manifolds

Let (M,g) be a smooth compact Riemannian manifold without boundary of dimension n>=6. We prove that {align*} \|u\|_{L^{2^*}(M,g)}^2 \le K^2\int_M\{|\nabla_g u|^2+c(n)R_gu^2\}dv_g +A\|u\|_{L^{2n/(n+2)}(M,g)}^2, {align*} for all u\in H^1(M), where 2^*=2n/(n-2), c(n)=(n-2)/[4(n-1)], R_g is the scalar curvature, $K^{-1}=\inf\|\nabla u\|_{L^2(\R^n)}\|u\|_{L^{2n/(n-2)}(\R^n)}^{-1}$ and A>0 is a constant depending on (M,g) only. The inequality is {\em sharp} in the sense that on any (M,g), $K$ can not be replaced by any smaller number and R_g can not be replaced by any continuous function which is smaller than R_g at some point. If (M,g) is not locally conformally flat, the exponent 2n/(n+2) can not be replaced by any smaller number. If (M,g) is locally conformally flat, a stronger inequality, with 2n/(n+2) replaced by 1, holds in all dimensions n>=3.Comment: 35 page

Blowup behavior for a degenerate elliptic sinh-Poisson equation with variable intensities

In this paper, we provide a complete blow-up picture for solution sequences to an elliptic sinh-Poisson equation with variable intensities arising in the context of the statistical mechanics description of two-dimensional turbulence, as initiated by Onsager. The vortex intensities are described in terms of a probability measure defined on the interval. Under Dirichlet boundary conditions we establish the exclusion of boundary blowup points, we show that the concentration mass does not have residual L1-terms and we determine the location of blowup points in terms of Kirchhoff's Hamiltonian. We allow the measure to be a general Borel measure, which could be "degenerate." Our main results are new for the standard sinh-Poisson equation as well