  The purpose of this paper is to propose methods for verifying the positivity
of a weak solution $ u $ of an elliptic problem assuming $ H^1_0 $-error
estimation $ \left\|u-\hat{u}\right\|_{H_{0}^{1}} \leq \rho $ given some
numerical approximation $ \hat{u} $ and an explicit error bound $ \rho $. We
provide a sufficient condition for the solution to be positive and analyze the
range of application of our method for elliptic problems with polynomial
nonlinearities. We present numerical examples where our method is applied to
some important problems.
