On scalar field equations with critical nonlinearity

Abstract

The paper concerns existence of solutions to the scalar field equation -Delta u = f(u), u > 0 in R-N, u is an element of D-1,D-2(R-N), N > 2, (0.1) when the nonlinearity f(s) is of the critical magnitude O(vertical bar s vertical bar((N+2)/(N-2))). A necessary existence condition is that the nonlinearity F(s) = integral(s) f divided by the "critical stem" expression vertical bar s vertical bar((N+2)/(N-2)) is either a constant or a non-monotone 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.