  We consider a class of nonlinear elliptic problems associated with models in
biophysics, which are described by the Poisson-Boltzmann equation (PBE). We
prove mathematical correctness of the problem, study a suitable class of
approximations, and deduce guaranteed and fully computable bounds of
approximation errors. The latter goal is achieved by means of the approach
suggested in [S. Repin, A posteriori error estimation for variational problems
with uniformly convex functionals. Math. Comp., 69:481-500, 2000] for convex
variational problems. Moreover, we establish the error identity, which defines
the error measure natural for the considered class of problems and show that it
yields computable majorants and minorants of the global error as well as
indicators of local errors that provide efficient adaptation of meshes.
Theoretical results are confirmed by a collection of numerical tests that
includes problems on $2D$ and $3D$ Lipschitz domains.
