2014/04/07 by Salvador Villegas, Villegas, Salvador
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP
paper · pdf · doi:10.48550/arxiv.1404.1722
15 pages
arxiv created 2014/04/07 · arxiv updated 2014/04/08
This paper is devoted to the study of stable radial solutions of -Δu=f(u) in ℝN∖ B1=\ x∈ ℝN : \vert x\vert≥ 1\, where f∈ C1(ℝ) and N≥ 2. We prove that such solutions are either large [in the sense that \vert u(r)\vert ≥ M r-N/2+√(N-1)+2 , if 2≤ N≤ 9; \vert u(r)\vert ≥ M log (r) , if N=10; \vert u(r)-u_∞ \vert ≥ M r\-N/2+√(N-1)+2 , if N≥ 11; ∀ r≥ r0, for some M>0, r0≥ 1] or small [in the sense that \vert u(r)\vert ≤ Mlog (r) , if N=2; \vert u(r)-u_∞ \vert ≤ M r\-N/2-√(N-1)+2; if N≥ 3; ∀ r≥ 2, for some M>0], where u_∞=limr→ ∞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)=log (r), ∀ r≥ 1, where α∈ ℝ ∖ \ 0\.