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LetN  ≥  2{N\geq 2}    and1  &lt;  p  &lt;    (   N  +  2   )   /    (   N  -  2   )   +{1&lt;p&lt;(N+2)/(N-2)_{+}}   . Consider the Lane–Emden equationΔ  ⁢  u   +   u  p    =  0{\Delta u+u^{p}=0}    inℝ  N{\mathbb{R}^{N}}    and recall the classical Liouville type theorem: if  u  is a non-negative classical solution of the Lane–Emden equation, thenu  ≡  0{u\equiv 0}   .

A Refined Approach for Non-Negative Entire Solutions of Δ u + u p = 0 with Subcritical Sobolev Growth