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Suppose $f \in L^2(\Omega)$ where $\Omega$ is bounded. The problem: for $a \in \mathbb{R}$ find $u_a \in H^1_0(\Omega)$ s.t

$$-\Delta u_a + \frac{m(u_a)}{a} = f$$

where $m(r) = \begin{cases} r &\text{if }r \leq 0, \\ 0 &\text{ otherwise} \end{cases}$ (and $u_a$ is a solution of the pde that depends on the number $a$).

My questions: 1) how do I go about showing that the solution exists and is unique? I can't use Lax-Milgram as $m$ is non-linear.

2) Suppose the solution exists. I am asked to prove that the solution $u_a \to u$, where $u \in K$ solves the variational inequality $$ \int_\Omega{\nabla u \cdot (\nabla v - \nabla u)} \geq \int_\Omega{f(v-u)}$$ for all $v \in K$ with $K = \{v \in H_0^1(\Omega) : v \geq 0 \text{ a.e.}\}$

For 2) I think maybe taking $a \to \infty$ is useful but I'm not sure really.

Help on either part would be appreciated. As this is homework, please don't give me the answer either. Thanks.

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Maybe you should split the solution $u_a$ into positive and negative part. Then you will have two linear problems. –  Beni Bogosel Apr 13 '12 at 10:50
    
Thanks. But I solved the problem using super and sub solutions. –  Court Apr 17 '12 at 21:09

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