The blow up analysis of solutions of the elliptic sinh-Gordon equation

Abstract

In this paper, using a geometric method we show that the blow-up values of the elliptic sinh-Gordon equation are multiples of 8π.

Changing-sign bubble solutions for an anisotropic sinh-Poisson equation

“…It is quite surprising that such accumulation of bubbles can occur for anisotropic sinh-Poisson equation with Dirichlet boundary condition. Our result is different from that in [9,15]. In [15] the authors showed that if the solutions concentrate positively and negatively at a same point, then the relation (3) must hold; And in [9] for a Neumann problem, a solution was constructed by superposing a positive and three negative bubbles centered near the origin.…”

“…Problem (2) relates to various dynamics of vorticity with respect to geophysical flows, rotating and stratified fluids and fluid layers excited by electromagnetic forces (see [10,14,16] and the references therein) and constant mean curvature surfaces studied by many works, see [15,19,23] and the references therein. Recently, the asymptotic behavior of solutions to (2) has been studied on a closed Riemann surface in [17] and [15] and the authors applied so-called "symmetrization method" and "Pohozaev identity" respectively to show that there possibly exist two different types of blow-up for a family of solutions to (2). Moreover, in [15] …”

“…Recently, the asymptotic behavior of solutions to (2) has been studied on a closed Riemann surface in [17] and [15] and the authors applied so-called "symmetrization method" and "Pohozaev identity" respectively to show that there possibly exist two different types of blow-up for a family of solutions to (2). Moreover, in [15] …”

“…Next, we have the following result. (15)) is a C 0 -stable critical point for the function F defined in (6). Then, problem (1) has a family of solutions u ε such that as ε → 0,…”

Non-simple blow-up solutions for the Neumann two-dimensional sinh-Gordon equation

“…In a recent paper [8], Jost, Wang, Ye and Zhou proved that a quantization of the limiting masses holds: m ± ( p) are multiples of 8π. It is the analogue of a result by Li and Shafrir [9] for the mean field equation.…”