Classical equations of Euclidean field theory

A study is made of the nonlinear elliptic equation $\Delta u=F(u)+f(x)$ in the whole of the space $R^n$. It is shown that if the function $f(x)$ has compact support and $F(u)$ satisfies the conditions $F(0)=0$, $F'(u)\geqslant\varkappa^2>0$, where $\varkappa$ is a constant, then a classical solution of this equation in the class of bounded functions exists, is unique, and decreases exponentially at infinity. Some cases when the condition $F'(u)\geqslant\varkappa^2$ is not satisfied are also considered. In particular, it is shown for the Goldstone model that at least two bounded solutions exist.