This is problem 8.11 in Evans.
Let $U\subset\mathbb{R}^n$ be open and bounded, with smooth boundary. Assume $\beta:\mathbb{R}\to\mathbb{R}$ is smooth with $0<a\leq\beta'\leq b$ for constants $a,b$. Let $f\in L^2(U)$. Formulate what it means for $u\in H^1(U)$ to be weak solution of the nonlinear BVP $$-\Delta u=f\text{ in }U\;;\quad\partial_{\nu}u+\beta(u)=0\text{ on }\partial U\qquad(\ast)$$ where $\nu$ is the outward normal. Prove that there exists a unique weak solution to $(\ast)$ in $H^1(U)$.
A quick integration by parts shows that $u\in H^1(U)$ is a weak solution of $(\ast)$ if $$\int_U\nabla v\cdot\nabla u+\int_{\partial U}v\beta(u)=\int_Ufv$$ for all $v\in H^1(U)$. Equivalently, $u$ should be a minimiser of the functional $$E(u)=\int_U\frac{1}{2}\|\nabla u\|^2-\int_Ufu+\int_{\partial U}B(u)$$ where $B'=\beta$. From here, I'd like to show that this energy is coercive and convex on (maybe some appropriate subspace of) $H^1(U)$. The conditions on $\beta'$ imply on the one hand that $B$ is strongly convex and w.l.o.g. is positive, and on the other that we can say things like $|\beta(u)-\beta(v)|\leq b|u-v|$, which "should" lead to some kind of continuity for the energy; facts like these are precisely what I need, but I can't upgrade them from heuristics to estimates. For one, it seems that $E$ is generally not bounded below on $H^1(U)$ (e.g. if $U$ is a disc of radius 3 in $\mathbb{R}^2$, $f\equiv1$ and $\beta(x)=x$, then $u_n\equiv n$ has $E(u_n)\to-\infty$). So I suppose it is necessary to restrict the domain of $E$, but I can't see how to do this in a useful way either. It is also possible that I have formulated the problem incorrectly from the start.
I'd appreciate very much some guidance on how to proceed with this problem.
