Variation d'un point de retournement par rapport au domaine

Abstract

Let Ω be a bounded and regular domain of ℝ^N, and Γ be its boundary.

For positive λ we consider the problem

$$(0.1)_\lambda \left\{ {\begin{array}{*{20}c}{ - \Delta u = \lambda e^u } & {in \Omega } \\{u = 0} & {on \Gamma } \\\end{array} } \right.$$

.

There exists a maximum value λ* of the parameter λ with 0<λ*<+∞, such that (0.1)_λ has at least one solution u in H ¹₀ (Ω) ∩ L^∞ (Ω), for λ ≥ [0,λ*[. Moreover, if the dimension N is less than 10, there exists a unique solution u* ε H ¹₀ (Ω) ∩ L^∞ (Ω) of problem (0.1)_λ*, and the point (λ*,u*) is then a turning point.

In this paper, we study the variation of this turning point with respect to the open set Ω, and more precisely we give an expression of the derivative of the turning point with respect to Ω (in a sense which is correctly defined in def. 2.1).

In problem (0.1)_λ we could have considered more general 2nd order elliptic operators and other types of positive increasing and convex nonlinearities, but for simplicity's sake we shall restrict ourselves to the particular problem stated above.