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We generalize in this article the classical Sobolev's and Sobolev's traceinequalities on the Grand Lebesgue Spaces under monomial weight instead theclassical Lebesgue or grand Lebesgue Spaces.  We will distinguish the classical Sobolev's inequality and the so-calledtrace Sobolev's inequality.

Exact constant in Sobolev's and Sobolev's trace inequalities for Grand Lebesgue Spaces with monomial weight