We give the precise behaviour of some solutions of a nonlinearelliptic B.V.P. in a bounded domain when a parameter approaches aneigenvalue of the principal part. If the nonlinearity has someregularity and the domain is for example convex, we also prove anonlinear version of Courant's Nodal theorem.

We consider the spatial behavior of the velocity field u(x, t) of a fluid filling the whole space $\xR^n$ ($n\ge2$) for arbitrarily small values of the time variable. We improve previous results on the spatial spreading by deducing the necessary conditions $\int u_h(x,t)u_k(x,t)\,{\rm d}x=c(t)\delta_{h,k}$ under more general assumptions on the localization of u. We also give some new examples of solutions which have a stronger spatial localization than in the generic case.