For n>=5 and 1<alpha<(n+4)/(n-4), the equation Delta^2 u = u^alpha admits no positive C^4 solution on complete noncompact manifolds with nonnegative Ricci curvature.
Case, The Obata--V\'etois argument and its applications, J
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Liouville theorem of the subcritical biharmonic equation on complete manifolds
For n>=5 and 1<alpha<(n+4)/(n-4), the equation Delta^2 u = u^alpha admits no positive C^4 solution on complete noncompact manifolds with nonnegative Ricci curvature.