Global analysis for a two-dimensional elliptic eigenvalue problem with the exponential nonlinearity

Abstract

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Convergence for a Liouville equation

“…In this case one has some good understanding (see [5]). In particular, when Ω = B 1 (0) and λ < 8π , there is a unique solution and as λ → 8π − , the behavior of solution set is analyzed in [19] and [23]; while λ ≥ 8π , it is known by using a Pohozaev-type identity that (1.1-2) admits no solution at all (see [20]). However, S.S.Lin [14] (see also [5], [23] and [24] for more) proved that if Ω is a suitable annulus, there are both radial and non-radial solutions for any λ > 0 .…”

“…As a side-remark, we point out that following [10] and [28], Chen and Li ([7] and [8]) used the moving plane method to obtain some beautiful apriori estimates for a class of elliptic partial differential equations in 2-dimensional domains. We remark that the special and important case when K = 1 was treated by K.Nagasaki and T.Suzuki (see [23] and [19]) using another method, which may also be applied to the case where K = constant (see [27]). It is clear that our result is also true if Ω is replaced by a compact 2-dimensional Riemannian manifold (M, g) and the equation is replaced by…”

Parabolic system of chemotaxis: Blowup in a finite and the infinite time

“…Among others are the application of the complex function theory to the blowup analysis of the family of solutions ( [19]), and the use of the rearrangement technique relative to a round sphere for spectral analysis of the linearized operator ( [24]). Consequently, we found that the set of stationary solutions C = {(A, V)} of (1.4) is much richer than the suspected, and some members are taking significant roles in the nonstationary problem.…”

Vortex solutions of Euler equation and its properties

“…Это обусловлено существенной неаналитичностью структуры максимaльного решения уравнения Лиувилля [29].…”