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$\\mathbb{R}^N$, $N\\geq 3$, $2^{*}=\\frac{2N}{N-2}$ is the critical Sobolev exponent and $\\lambda>0$ a positive parameter.\n  The main result of the paper shows that if $N=4,5,6$ and $\\lambda$ is close to zero there are no sign-changing solutions of the form $$u_\\lambda=PU_{\\delta_1,\\xi}-PU_{\\delta_2,\\xi}+w_\\lambda, $$ where $PU_{\\delta_i}$ is the projection 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