The following problem comes from:


12.9 Exercises

12.3: Let $\Omega$ be a bounded domain in the plane with smooth boundary. Let $f$ be a positive, strictly convex function over the reals. Let $w\in L^2(\Omega)$. Show that there exists a unique $u\in H^1(\Omega)$ that is a weak solution to the equation $$ \begin{cases} -\Delta u+f'(u)=w\quad \text{in} \quad \Omega, \\ u=0 \quad \text{on} \quad \partial\Omega. \\ \end{cases} $$ Do you need additional assumptions on $f$ ?

Suppose that the assumption on the positivity and convexity of $f=f(u)$ for all $u\in\mathbb{R}$ in the problem is replaced by assuming that $f=f(u)$ is only defined for $u\in [a,b]$, is positive and convex for $u\in [a,b]$, where $a$, $b\in\mathbb{R}$ are finite given numbers. The same conclusion remains true?


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