Case $d=4$:

The result is perfectly regular, and is simply given by

$$J_{0} = V_{0} \left( m_{\mathfrak{N}}^{2} - 2\lambda\, V_{0}^{2} \right).$$

Not much can be concluded in this case, though. Of course we must not forget that both $V_{0}$ and $m_{\mathfrak{N}}$ are functions of $\lambda$ and $J_{0}$, in such a way that the right-hand side of this equation remains positive for positive $J_{0}$.

Although this equation has the general form expected for an interacting theory, it is crucial to note that the sign of the second term is reversed. Since we must have $\lambda>0$, this term is necessarily negative. This is not really all that surprising, for one must not forget that it is not to be expected that this calculational technique can produce predictions about $\lambda_{R}$, which is a parameter related to the fourth moment of the distribution, that is of course absent from a Gaussian approximation. The reversed sign, that seems to indicate that increasing $\lambda$ works in the way opposite to what one would expect, appears there as a warning about this limitation. Of course, interpreting this term as a prediction for $\lambda_{R}$ would be absurd, since it would imply that the renormalized coupling constant is negative, and thus would correspond to an unstable renormalized model.