Global analysis for a two-dimensional elliptic eigenvalue problem with the exponential nonlinearity. (English) Zbl 0785.35045

Suzuki, Takashi

Ann. Inst. Henri Poincaré, Anal. Non Linéaire 9, No. 4, 367-398 (1992).

Summary: For the Emden-Fowler equation \(-\Delta u=\lambda e^ u\) in \(\Omega\subset\mathbb{R}^ 2\), the connectivity of the trivial solution and the one-point blow-up singular limit is studied with respect to the parameter \(\lambda>0\). The connectivity is assured when the domain \(\Omega\) is simply connected and the total mass \(\sum=\int_ \Omega\lambda e^ udx\) tends to \(8\pi\) from below, which is a generalization for the case that \(\Omega\) is a ball.

MSC:

35J65	Nonlinear boundary value problems for linear elliptic equations
47J10	Nonlinear spectral theory, nonlinear eigenvalue problems
35B30	Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35J60	Nonlinear elliptic equations

Keywords:

global bifurcation; rearrangement; bounded domain; nonlinear elliptic eigenvalue problem; Emden-Fowler equation; one-point blow-up singular limit

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