We construct explicit solutions of a number of Stieltjes moment problems based on moments of the form ${rho}_{1}^{(r)}(n)=(2rn)!$ and ${rho}_{2}^{(r)}(n)=[(rn)!]^{2}$, $r=1,2,...$, $n=0,1,2,...$, textit{i.e.} we find functions $W^{(r)}_{1,2}(x)>0$ satisfying $int_{0}^{infty}x^{n}W^{(r)}_{1,2}(x)dx = {rho}_{1,2}^{(r)}(n)$. It is shown using criteria for uniqueness and non-uniqueness (Carleman, Krein, Berg, Pakes, Stoyanov) that for $r>1$ both ${rho}_{1,2}^{(r)}(n)$ give rise to non-unique solutions. Examples of such solutions are constructed using the technique of the inverse Mellin transform supplemented by a Mellin convolution. We outline a general method of generating non-unique solutions for moment problems generalizing ${rho}_{1,2}^{(r)}(n)$, such as the product ${rho}_{1}^{(r)}(n)cdot{rho}_{2}^{(r)}(n)$ and $[(rn)!]^{p}$, $p=3,4,...$.
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