# Application of Schauder fixed point theorem

Let $\Omega$ an open bounded in $\mathbb{R}^n$, and let $A(x,u)$ an Caratheodory function, bounded and coercive, i.e., $$|A(x,u)|\leq \beta$$ $$A(x,u) \geq \alpha I,\quad \alpha > 0$$ and let $g(x,u) u \geq 0,\quad \forall u \in \mathbb{R},\quad \mbox{a.e. } x \in \Omega$ $|g(x,u)|\leq h_t(x),\quad \forall u, |u|\leq t,\quad h_t \in L^1(\Omega), \quad \forall t > 0$ fixed. Let $\epsilon > 0$, and let $$g_{\epsilon}(x,s)=\dfrac{g(x,s)}{1+\epsilon |g(x,s)|}$$ where $g_{\epsilon}$ is Caratheodory function such that $|g_{\epsilon}|\leq |g|$ and $|g_{\epsilon}|\leq \dfrac{1}{\epsilon}$.

Let $f \in H^{-1}(\Omega)$. We consider the problem $$\begin{cases} & - \mathrm{div} A(x,u^{\epsilon}) \nabla u^{\epsilon} + g_{\epsilon} (x,u^{\epsilon})=f\\ & u^{\epsilon} \in H^1_0(\Omega) \end{cases}$$ How we can prove with the Schauder fixed point theorem that this problem admits one last solution $u_{\epsilon} \in H^1_0(\Omega)$? Have you an book or a paper? Thanks for the help.

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