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It is known that (0.1) admits infinitely many radially symmetric entire solutions. These solutions may have either a (negative) logarithmic behaviour or a (negative) quadratic behaviour at infinity. Up to translations, we know that there is only one radial entire solution with the former behaviour, which is called ‘maximal radial entire solution’, and infinitely many radial entire solutions with the latter behaviour, which are called ‘non-maximal radial entire solutions’. The necessary and sufficient conditions for an entire solution u of (0.1) to be the maximal radial entire solution are presented in [7] recently. In this paper, we will give the necessary and sufficient conditions for an entire solution u of (0.1) to be a non-maximal radial entire solution.
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