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By Conti M., Terracini S., Verzini G.

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Dancer, Competing species systems with diffusion and large interaction, Rend. Sem. Mat. Fis. N. H. Du, Competing species equations with diffusion, large interactions, and jumping nonlinearities, J. N. M. Guo, Uniqueness and stability for solutions of competing species equations with large interactions, Comm. Appl. Nonlinear Anal. N. Dancer, D. Hilhorst, M. A. Peletier, Spatial segregation limit of a competition– diffusion system, European J. Appl. Math. 10 (1999), 97–115 32 [12] P. Hartman, A. Winter, On the local behaviour of solutions of non–parabolic partial differenzial equations (III) Approximation by spherical harmonics, Amer.

Mat. Fis. N. H. Du, Competing species equations with diffusion, large interactions, and jumping nonlinearities, J. N. M. Guo, Uniqueness and stability for solutions of competing species equations with large interactions, Comm. Appl. Nonlinear Anal. N. Dancer, D. Hilhorst, M. A. Peletier, Spatial segregation limit of a competition– diffusion system, European J. Appl. Math. 10 (1999), 97–115 32 [12] P. Hartman, A. Winter, On the local behaviour of solutions of non–parabolic partial differenzial equations (III) Approximation by spherical harmonics, Amer.

2 it suffices to show that, for i = j, u0i and u0j have disjoint support: this is consequence of property (iii) and of the strong convergence of U ε to its weak limit. e. x ∈ Ω and proves supp(u0i ) ∩ supp(u0j ) = ∅ for every i = j. Moreover by (iv) we also obtain J(U 0 ) = c0 , finally proving the assertion. As a final remark, note that, if we pass to the limit in (17) as ε → 0, by exploiting the strong convergence of the solutions U ε to the k–uple U 0 and (iii), we realize that −∆u0i (x) = f (u0i (x)) x ∈ supp(u0i ) ∀i.

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