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.