On the Nonlinear Eigenvalue Problem Δu + λe u = 0

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

“…Our purpose is to show that this phenomenon holds in more general situations. We can prove the following theorem, which is a refinement of our previous work [ 18] : THEOREM 2. -Let S2 be simply connected.…”

“…Propositions 1 and 2 are known when p is real analytic. For instance, see [18], Proposition 2 and [3], p. 108, respectively. That is enough for showing Lemmas 1 and 3.…”

“…In fact, holds for some ~==~(Y)e(0, + oo). We furthermore have and See [18] for the proof, for instance.…”

Supercritical Mean Field Equations on Convex Domains and the Onsager’s Statistical Description of Two-Dimensional Turbulence

“…If not, under which conditions (if any) the entropy has only one inflection point? In particular, is it true that the global branch of solutions of P (λ, Ω ρ ) with ρ small enough has just one bending point, no bifurcation points and it is connected with the blow-up solution's branch as λ ց (8π) + (as for Q(µ, Ω) on nearly circular domains [68])? Can we answer this question at least on convex, regular and symmetric domains?…”

“…This rules out the possibility of having more than one single peak blow-up solution. So far, it seems that in particular the global connectivity of the solution's branch is known only for domains which are close in C 2 -norm to a disk, see [68]. Of course, if (say in case Ω = Ω ρ with ρ small enough) the entropy really has just one inflection point, then it will coincide with the point on the continuation of G ρ,1 where the first eigenvalue of the linearized problem for P (λ, Ω ρ ) will finally vanish.…”