Exhaustive existence and non-existence results for some prototype polyharmonic equations in the whole space

International audienceIn 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 nonlinearity ∆^m u = ±u^α in R n 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 R n. Finally, we identify all the situations where any entire non-trivial, non-negative classical solution must be positive everywhere