Existence, decay and symmetry of solutions of hamiltonian systems in ℝ^{N}

Boyan Sirakov Laboratoire d’Analyse Numérique Université Paris 6 Tour 55-65, 5e étage 75252 Paris Cedex, France

                 sirakov@ann.jussieu.fr

We study the existence and the decay of solutions of systems of the type

| \begin{matrix}−Δu + u & = & g(r, v) \\ −Δv + v & = & f(r, u) \end{matrix}  (1)

in ℝ^{N}, N ≥ 3. Using an infinite dimensional linking theorem we show that such a system has a positive radial solution under the standard “mountain pass” assumptions on the nonlinearities, provided their growths at infinity lie below the so-called “critical hyperbola”. This result generalises a recent paper by de Figueiredo and Yang ([1]). We also prove their conjecture that the solutions decay at infinity together with their derivatives.

We also study the symmetry of solutions of more general systems in ℝ^{N}

\{ \begin{matrix}Δu_{i} + f_{i}(r, u_{1}, ⋯, u_{n}) = 0 & in ℝ^{N}, i = 1, ⋯, n, \\ u_{i} > 0 & in ℝ^{N}, \\ u_{i}(x) → 0 & as r = |x| → ∞, \end{matrix}  (2)

where n ≥ 1, N ≥ 2 are arbitrary integers. We prove that solutions of (2) are necessarily radial with respect to the same origin provided the system is cooperative, fully coupled and all n-principal minors of (\frac{∂f_{i}}{∂u_{j}}) are nonnegative near zero. This is an extension to ℝ^{N} of a well-known result of Troy ([3]) and a generalisation to systems of the symmetry results for single equations ([2]). This result is obtained in a joint work with J. Busca.

References

[1] D.G. de Figueiredo, J. Yang, Decay, symmetry and existence of positive solutions of semilinear elliptic systems, To Appear in Nonl. Anal.

[2] Y. Li, W.-M. Ni, Radial symmetry of positive solutions of nonlinear elliptic equations, Comm. Partial Diff. Eq. 18 (1993), pp. 1043-1054.

[3] W.C. Troy, Symmetry properties in systems of semilinear elliptic equations, J. Diff. Eq. 42 (1981), pp. 400-413.