Essential goods: How does one restrict the utility function?
In order to have $x^*_j&gt;0$, it is possible to impose either
(i) a condition on the marginal utility: $\lim_{x_j\rightarrow0} \partial{U}/\partial{x_j}(x)=+\infty$
(ii) or an inequality $x_j \geq a_j&gt;0$ where $a_j$ is interpreted as a subsistence level of $x_j$
Often the utility function is reparameterized and written $U(x-a)$ with the constraint $X:=x-a\geq 0$, which ensures that $x^*\geq a$ at the inner optimum.
Example: in the Cobb-Douglas case with subsistence levels, we have
$$ U(X_1,X_2)=U(x_1-a_1,x_2-a_2)=(x_1-a_1)^\alpha(x_2-a_2)^\beta $$
Provided that the income $m$ is high enough, the Marshallian demands are given by:
$$x^*_1 = a_1 + \frac{\alpha}{\alpha+\beta}\frac{m-p_1a_1-p_2a_2}{p_1}\\
x^*_2 = a_2 + \frac{\beta}{\alpha+\beta}\frac{m-p_1a_1-p_2a_2}{p_2}. $$