# Existence of a solution for the Laplace equation with sub-linear non-linearity

At first, I do apologize if my question is silly. I know that by variational methods it is possible to prove the existence of a solution for $$\begin{cases} -\Delta u = u^p & \Omega \subset \mathbb{R}^n \ u=0 & \partial \Omega \end{cases}$$ for $$1 < p < \frac{n+2}{n-2}$$.

But I don’t have any idea about the sublinear case, namely, $$p<1$$. Also, I don’t have any information about the case $$p<0$$, too.