On a semilinear elliptic equation from the sigma model of quantum field theory

In this paper, we consider both the existence and the non-existence of solutions to the generalized semilinear elliptic equations from the sigma model of quantum field theory of condense matter. The interesting part of the result is that the conditions to prove the existence and the non-existence of the solution are not determined by the nonlinearity of the nonlinear function but rather the quantitative value of its source term.