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

Abstract

For the Neumann sinh-Gordon equation on the unit ball B ⊂R^2\ud $-\Delta u = λ^+(e^u/\int_B e^u-1/\pi)-λ^-(e^{-u}/\int_B e^{-u}-1/\pi)$ in $B$, $∂_ν=0$ on $\partial B$ we construct sequence of solutions which exhibit a multiple blow up at the origin, where λ± are positive parameters. It answers partially an open problem formulated in Jost et al. [Calc Var Partial Diff Equ 31(2):263–276]

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.…”

Analytic aspects of the Tzitzéica equation: blow-up analysis and existence results

“…Later, in [24] the case of general h 1 , h 2 was studied and the authors proved an analogous quantization property as the one in (3), namely that the blow-up limits are multiple of 8π. The latter blow-up situation may indeed occur, see [14] and [17].…”

Blow-up analysis for an elliptic equation describing stationary vortex flows with variable intensities in 2D-turbulence

“…In this case, it was noticed in [20] that the blow-up masses satisfy a quadratic identity. See also [13,8] for further results in this direction. From such a property, an improved sharp Trudinger-Moser inequality was derived.…”

“…(7) is the neutral mean field equation derived in [14,22]. The variational functionals associated to (1) and (5), on the other hand, are given by 1] log Ω e αv dx P(dα) (8) and…”