On scalar field equations with critical nonlinearity

The paper concerns existence of solutions to the scalar field equation (0.1) $$-\triangle u = f(u),\quad u > 0\,\,\rm{in}\,\,\mathbb{R}^{N},\quad u\,\in\,\mathcal{D}^{1,2}(\mathbb{R}^{N}), N > 2,$$ when the nonlinearity f(s) is of the critical magnitude $$O(|{s}|^{(N+2)/(N-2)})$$ . A necessary existence condition is that the nonlinearity $$F(s) = \int^s$$ f divided by the “critical stem” expression $$|{s}|^{(N+2)/(N-2)}$$ is either a constant or a nonmonotone function. Two sufficient conditions known in the literature are: the nonlinearity has the form of a critical stem with a positive perturbation (Lions), and the nonlinearity has selfsimilar oscillations ([11]). Existence in this paper is proved also when the nonlinearity has the form of the stem with a sufficiently small negative perturbation, of the stem with a negative perturbation of sufficiently fast decay rate (but not pointwise small), or of the stem with a perturbation with sufficiently large positive part.