Abstract
We study and obtain formulas for the asymptotic behavior as ¦x¦→∞ of C 2 solutions of the semilinear equation Δu=f(x, u), xεΩ (*) where Ω is the complement of some ball in ℝn and f is continuous and nonlinear in u. If, for large x, f is nearly radially symmetric in x, we give conditions under which each positive solution of (*) is asymptotic, as ¦x¦→∞, to some radially symmetric function. Our results can also be useful when f is only bounded above or below by a function which is radially symmetric in x or when the solution oscillates in sign. Examples when f has power-like growth or exponential growth in the variables x and u usefully illustrate our results.
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Communicated by J. Serrin
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Taliaferro, S.D. Asymptotic behavior of solutions of nonlinear elliptic equations. Arch. Rational Mech. Anal. 122, 105–121 (1993). https://doi.org/10.1007/BF00378163
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DOI: https://doi.org/10.1007/BF00378163