  The paper deals with minimum energy problems in the presence of external
fields on a locally compact space $X$ with respect to a function kernel
$\kappa$ satisfying the energy and consistency principles. For quite a general
(not necessarily lower semicontinuous) external field $f$, we establish
sufficient and/or necessary conditions for the existence of $\lambda_{A,f}$
minimizing the Gauss functional \[\int\kappa(x,y)\,d(\mu\otimes\mu)(x,y)+2\int
f\,d\mu\] over all positive Radon measures $\mu$ with $\mu(X)=1$, concentrated
on quite a general (not necessarily closed or bounded) $A\subset X$, thereby
giving an answer to a question raised by M. Ohtsuka (J. Sci. Hiroshima Univ.,
1961). Such results are specified for the Riesz kernels $|x-y|^{\alpha-n}$,
$0<\alpha<n$, on $\mathbb R^n$, $n\geqslant2$, and are illustrated by some
examples. Furthermore, we provide various alternative characterizations of the
minimizer $\lambda_{A,f}$, and as a by-product we analyze the strong and vague
continuity of $\lambda_{A,f}$ under the exhaustion of $A$ by compact $K\subset
A$. The results obtained hold true and are new for many interesting kernels in
classical and modern potential theory.
