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 ?

Thanks in advance


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