89 research outputs found
Mountain pass solutions for a mean field equation from two-dimensional turbulence
Using Struwe's "monotonicity trick" and the recent blow-up analysis of
Ohtsuka and Suzuki, we prove the existence of mountain pass solutions to a mean
field equation arising in two-dimensional turbulence.Comment: 13 page
On a nonlinear elliptic system from Maxwell-Chern-Simons vortex theory
We define an abstract nonlinear elliptic system, admitting a variational
structure, and including the vortex equations for some Maxwell-Chern-Simons
gauge theories as special cases. We analyze the asymptotic behavior of its
solutions, and we provide a general simplified framework for the asymptotics
previously derived in those special cases. As a byproduct of our abstract
formulation, we also find some new qualitative properties of solutions
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
Sign-changing two-peak solutions for an elliptic free boundary problem related to confined plasmas
By a perturbative argument, we construct solutions for a plasma-type problem
with two opposite-signed sharp peaks at levels and , respectively,
where . We establish some physically relevant qualitative
properties for such solutions, including the connectedness of the level sets
and the asymptotic location of the peaks as .Comment: 22 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, and A>0 is a constant depending
on (M,g) only. The inequality is {\em sharp} in the sense that on any (M,g),
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
- …