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We prove that the Dirichlet problem for the Lane-Emden equation in ahalf-space has no positive solutions which grow at most like the distance tothe boundary to a power given by the natural scaling exponent of the equation;in other words, we rule out {\it type~I grow-up} solutions. Such a nonexistenceresult was previously available only for bounded solutions, or under arestriction on the power in the nonlinearity. Instrumental in the proof arelocal pointwise bounds for the logarithmic gradient of the solution and itsnormal derivative, which we also establish.

Liouville-type theorems for unbounded solutions of elliptic equations in half-spaces