by (32) and the general solution $\Phi(u)$ is defined to be the concentration profile which minimizes the free energy functional (14). In the following, we discuss the properties of the general solution $\Phi(u)$ separately for the four regions distinguished in Fig. 2.

i\) In the delocalized case, the boundary conditions (37) occurring at infinity together with the differential equation (26) valid for the entire ($-\infty, \infty)$ range in $u$ are sufficient to determine the distribution $\Phi(u)$.

For the other cases, the boundary conditions (37) have to be supplemented by additional conditions at finite values of $u$; the distribution $\Phi(u)$ is described by (26) only in a finite interval of $u$.

ii\) In the case where $h(u)<0$ for all $u$, it follows from the requirement $\Phi(u) \geq 0$ that $\Phi(u)-h(u) >0$ and thus all possible solutions of (26) have strictly positive curvature as can be seen by looking at (26). The boundary conditions (37), which imply that $\Phi(u)=0$ as $u\rightarrow \pm \infty$, can not be satisfied for any non-vanishing solution of (26). Consequently, the protein distribution which minimizes the free energy is given identically by $\Phi(u)=0$. This vanishing solution was also found for $b=0$.

iii\) When $h(u)$ is positive in some 