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We prove existence of extremal functions for some Rellich-Sobolev typeinequalities involving the $L^{2}$ norm of the Laplacian as a leading term andthe $L^{2}$ norm of the gradient, weighted with a Hardy potential. Moreover weexhibit a breaking symmetry phenomenon when the nonlinearity has a growth closeto the critical one and the singular potential increases in strength.

Radial and non radial ground states for a class of dilation invariant fourth order semilinear elliptic equations on $\mathbb{R}^{n}$