Dichotomy of stable radial solutions of -Δu=f(u) outside a ball

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This paper is devoted to the study of stable radial solutions of -Δu=f(u) in Rⁿ B₁={ x∈ Rⁿ: x≥ 1}, where f∈ C¹(R) and N≥ 2. We prove that such solutions are either large [in the sense that u(r) ≥ M r-N/2+√N-1+2, if 2≤ N≤ 9; u(r) ≥ M (r), if N=10; u(r)-u∞ ≥ M rN/2+√N-1+2, if N≥ 11; ∀ r≥ r₀, for some M>0, r₀≥ 1] or small [in the sense that u(r) ≤ M (r), if N=2; u(r)-u∞ ≤ M rN/2-√N-1+2; if N≥ 3; ∀ r≥ 2, for some M>0], where u∞=ᵣ→ ∞u(r)∈ [-∞,+∞]. These results can be applied to stable outside a compact set radial solutions of equations of the type -Δu=g(u) in Rⁿ. We prove also the optimality of these results, by considering solutions of the form u(r)=rα or u(r)= (r), ∀ r≥ 1, where α∈ R { 0}.