Introduction and statement of the results

The principal objective of this work is the classification of the nonnegative

solutions of

(1) Au = u2

in a bounded and smooth domain D in

Rd.

Equation (1) is one of the simplest non-

linear elliptic partial differential equations. It belongs to a larger class of semilinear

(or quasilinear) partial differential equations of the type

(2) Lu = if;(u)

where L is a second order elliptic differential operator and ip is a convex nonnega-

tive increasing function. These equations with ip(x,u) =

k(x)ua,

k 0, a 1 are

connected to various problems in meteorology (Emden), theory of atomic spectra

(Thomas-Fermi), astrophysics (Chandrasekhar) or geometry (Yamabe problem).

An extensive bibliography about this subject is contained in [V96]. A lot of math-

ematicians have contributed to the theoretical study of this type of equations among

which Baras, Brezis, Keller, Loewner, Nirenberg, Osserman, Pierre and Strauss and

more recently, since 1990, Dynkin, Gmira, Kuznetsov, Le Gall, Marcus, Sheu and

Veron (see the bibliography at the end of this memoir). For technical reasons,

we restrict ourselves to equation (1). However, it seems plausible that our work

will serve as a good starting point towards the obtention of similar results for the

general equation (2). Even though many results we prove are purely analytic, we

extensively use probabilistic tools such as superprocesses and the Brownian snake.

The set of solutions. We denote by U the set of all nonnegative solutions

of

(3) Au =

4u2

in D,

where D is a bounded and smooth domain in Rd. The factor 4 here is for the

convenience of the probabilistic representation (see below) and has no analytic

meaning. The set U is compact for the topology of the uniform convergence on

the compact subsets of D (see Proposition 1.12) and is endowed with a semi-group

structure: if u, v G hi,

u 0 v := Sup{w; G U ; w ^ u -f- v}.

(See [DK98b] and Section 1.3.5).

Nonlinear trace theory. The basic problem is to represent all nonnegative

solutions of Lu — ip(u) in a domain D in terms of their trace on the boundary dD.

After the preliminary work of Gmira and Veron (1989-1991), a program for such a

description was initiated by Dynkin in 1990 and 1991 (see in particular [Dy93]).

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