(Y~I- ~2 )dx d = 0 that y = -~ ~s included between O and i. We have in the square Cnm. ) is equivalent to the one obtained in case ii) of the proposition 3. ), and more p a r t i c u l a r l y w i t h the stable solutions. ) u(t) + Au(t) u(O) = u ° - Xu(t) + P(u(t)) = O in given in H2(~) ~ H~(9) We discretise in time this e v o l u t i o n problem.

Will be given in section IV. w 32 IV. ]. ]. = ~ lu p - ~u q - m~ We have BR heorem classical equations (see state the result we be positive cons- : Under result , radial, decreasing (where (2) is a necessary results case. in r = [xl) : ~n to solve ~u q - mu > O. 2. l, ~ , m u @ C°°(~ n) for solving (of radius R) with Dirichlet the following ~ that of (I). More general R -~+ ~ . We - 0~ (I) in here is a kind of model but in a ball (I-R) We first m >O] D ~ u ( x ) ~ C e $ ~ m In this section we consider letting solution In [5] , we prove solution considered (|) we in non linear Schr~dinger o~ , let ~q+l q+l ~ (u(x) trivial of equation wares or non linear .

### Bifurcation and Nonlinear Eigenvalue Problems: Proceedings, Université de Paris XIII, Villetaneuse, France, October 2–4, 1978 by M. F. Barnsley (auth.), C. Bardos, J. M. Lasry, M. Schatzman (eds.)

by Jason

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