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On scalar field equations with critical nonlinearity

Authors:

Kyril Tintarev

Institution:

(1) Department of Mathematics, Uppsala University, SE-751 06 Uppsala, Sweden

Abstract:

The paper concerns existence of solutions to the scalar field equation

$-\triangle u = f(u),\quad u > 0\,\,\rm{in}\,\,\mathbb{R}^{N},\quad u\,\in\,\mathcal{D}^{1,2}(\mathbb{R}^{N}), N > 2,$

((0.1))

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. Dedicated to Felix Browder on the occasion of his 80-th birthday

Keywords:

Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) 35J20 35J60 49J35

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