Abstract
In this paper, we are interested in entire, non-trivial, non-negative solutions and/or entire positive solutions to the simplest models of polyharmonic equations with power-type nonlinearityΔmu=±uαin Rn with n⩾1, m⩾1, and α∈R. We aim to study the existence and non-existence of such classical solutions to the above equations in the full range of the constants n, m and α. Remarkably, we are able to provide necessary and sufficient conditions on the exponent α to guarantee the existence of such solutions in Rn. Finally, we identify all the situations where any entire non-trivial, non-negative classical solution must be positive everywhere.
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