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On scalar field equations with critical nonlinearity
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2008 (English)In: Journal of Fixed Point Theory and Applications, ISSN 1661-7738, Vol. 4, no 1, 97-106 p.Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
2008. Vol. 4, no 1, 97-106 p.
Keyword [en]
Semilinear elliptic equations, scalar field equation, concentration compactness, variational problems
National Category
URN: urn:nbn:se:uu:diva-137342DOI: 10.1007/s11784-007-0070-9ISI: 000263027900008OAI: oai:DiVA.org:uu-137342DiVA: diva2:378323
Available from: 2010-12-15 Created: 2010-12-15 Last updated: 2010-12-15Bibliographically approved

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