Minimum energy problems with external fields on locally compact spaces

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 $$\int \kappa (x,y)d(\mu \otimes \mu )(x,y)+2\int fd\mu$$ 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 $\mathrm{\mid }x-y{\mathrm{\mid }}^{\alpha -n}$, $0<\alpha <n$, on ${\mathbb{R}}^n$, $n⩾2$, 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.