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| We investigate the problem of entire solutions for a class of fourth order, dilation invariant, semilinear elliptic equations with power-type weights and with subcritical or critical growth in the nonlinear term. These equations define non compact variational problems and are characterized by the presence of a term containing lower order derivatives, whose strength is ruled by a parameter $\lambda$. We can prove existence of entire solutions found as extremal functions for some Rellich-Sobolev type inequalities. Moreover, when the nonlinearity is suitably close to the critical one and the parameter $\lambda$ is large, symmetry breaking phenomena occur, and the asymptotic behavior of radial and non radial ground states can be somehow described. As a by-product result we obtain information on the sign of ground state solutions. |
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