Olivier Rey

The role of the Green's function in a non-linear elliptic equation involving the critical Sobolev exponent

This paper is concerned with non-linear elliptic problems of the type (Pe): −Δu = u(N + 2)(N − 2) + eu, u > 0 on Ω; u = 0 on ∂Ω, where Ω is a smooth and bounded domain in RN, N ≥ 4, and e > 0. We show that if the ue are solutions of (Pe) which concentrate around a point as e → 0, then this point cannot be on the boundary of Ω and is a critical point of the regular part of the Greens function. Conversely, we show that for N ≥ 5 and any non-degenerate critical point x0 of the regular part of the Greens function, there exist solutions of (Pe) concentrating around x0 as e → 0.

Proof of two conjectures of H. Brezis and L.A. Peletier

AbstractIn this paper, we consider the problem: −Δu=N(N−2)up−ɛ, u>0 on Ω; u=0 on ∂Ω, where Ω is a smooth and bounded domain inRN, N≥3, p=

Concentration of solutions to elliptic equations with critical nonlinearity

\frac{{N + 2}}{{N - 2}}

Bifurcation from infinity in a nonlinear elliptic equation involving the limiting Sobolev exponent

, and ε>0. We prove a conjecture of H. Brezis and L.A. Peletier about the asymptotic behaviour of solutions of this problem which are minimizing for the Sobolev inequality as ε goes to zero. We give similar results concerning the related problem: −Δu=N(N−2)up+εu, u>0 on Ω; u=0 on ∂Ω, for N is larger than 4.

The question of interior blow-up points for an elliptic Neumann problem: the critical case

This paper is concerned with the nonlinear elliptic problem (Pe): -Δu = up+e, u > 0 in Ω; u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝn, n ≥ 3, p + 1 = 2n/(n - 2) is the critical Sobolev exponent and e is a small positive parameter. In contrast with the subcritical problem (P- e) studied by Han [11] and Rey [17], we show that (Pe) has no single peaked solution for small e.

Elliptic equations with limiting Sobolev exponent: the impact of the Green's function

Abstract We study the asymptotic behavior as e goes to zero of solutions in H 0 1 ( Ω ) to the equation: − Δ u = | u | 4 / ( N − 2 ) u + e f ( x ) , where Ω is a bounded domain in RN. We show the existence of solutions to the problem which blow-up at some well-defined points, depending on f, for e = 0.

On a variational problem with lack of compactness: the topological effect of the critical points at infinity

Let D = {z = (x, y) e R2/x 2 + y2 < 1 }, and H be a function from R 3 to R. If u: D ~ R a is the parametrization of a surface whose mean curvature at the point u(z) is H(u(z)), then u solves in D the equation Au=2H(u)UxA%. (1.1) We are interested here in the behavior of the heat flow corresponding to this equation, that is we look for f : [0, + oo[ x D ~ R 3 which satisfies ft = A f 2H( f ) f x A fy. (1.2) More precisely, boundary data z : O D ~ R a and initial data ~b:D~R a, with the compatibility condition

A multiplicity result for a variational problem with lack of compactness

10D = X, being given, we consider the problem