# Why does this elliptic quasilinear problem have a weak solution?

Let $\Omega \subset R^n$ a bounded domain with smooth boundary and $\beta \in (0,1)$ fixed. Let $\overline{U}$ the weak solution of the problem

$$\Delta \overline{U} = 1 \text{ in } \Omega \\ \overline{U} = 0 \text{ in } \partial \Omega .$$

It can be proved that this solution is bounded and positive in $\Omega$.

Now consider $\Gamma : [0,+ \infty) \rightarrow \mathbb R$. Fix $m \in (0,1 + \beta)$ and suppose that $\Gamma (t) \leq \Gamma_0(1 + t^m)$ for all $t \in \mathbb R,$ where $\Gamma_0$ is a fixed constant.

Let $\overline{U} : = u_0 .$ I am reading a paper and the authors say that :

Given $\epsilon$ the above problem admits a weak solution

$$\Delta u_1 + \frac{u_1}{(u_0 + \epsilon)^{1+\beta}} \Gamma(|\nabla u_1|) = 0\text{ in } \Omega \\ u_1 = 0 \text{ in } \partial \Omega .$$ and $u_0 \geq u_1$ in $\Omega.$

I dont have much experience in quasilinear problems. It seems that these things can be obtained by the use of general results concerning quasilienar equations, but I am not finding anything that helps me answering these questions. Someone could help me ?