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Generalized minimizer solutions for equations with the p-Laplacian and a potential term
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2008 (English)In: Proceedings of the Royal Society of Edinburgh. Section A Mathematics, ISSN 0308-2105, Vol. 138, no 1, 201-221 p.Article in journal (Refereed) Published
Abstract [en]

Let Omega be a domain in R-N (possibly unbounded), N >= 2, 1 < p < infinity, and let V is an element of L-loc(infinity)(Omega). Consider the energy functional Q(V) on C-c(infinity)(Omega) and its Gateaux derivative Q'(V), respectively, given by Q(V)(u) =(def) 1/p integral(Omega)(vertical bar del u vertical bar(p) + V vertical bar u vertical bar(p))dx, Q'(V)(u) = div(vertical bar del u vertical bar(p-2)del u vertical bar(p-2)u, for u is an element of C-c(infinity) (Omega). Assume that Q(V) > 0 on C-c(infinity) (Omega) \ {0} and that Q(V) does not have a ground state (in the sense of a null sequence for Q(V) that converges in L-loc(p)(Omega) to a positive function phi is an element of C-loc(1)(Omega), a ground state). Finally, let f is an element of D'(Omega) be such that the functional u -> Q(V) (u) - < u, f > : C-c(infinity)(Omega) -> R is bounded from below. Then the equation Q'(V)(u) = f has a solution u(0) W-loc(1,p)(Omega) in the sense of distributions. This solution also minimizes the functional u -> Q(V)** (u) - < u, f > : C-c(infinity) (Omega) -> R, where Q(V)** denotes the bipolar (Gamma-regularization) of Q(V) and Q(V)** is the largest convex, weakly lower sermcontinuous functional on C-c(infinity)(Omega) that satisfies Q(V)** <= Q(V). (The original energy functional Q(V) is not necessarily convex.).

Place, publisher, year, edition, pages
2008. Vol. 138, no 1, 201-221 p.
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URN: urn:nbn:se:uu:diva-112389DOI: 10.1017/S0308210506000904ISI: 000253780800010OAI: oai:DiVA.org:uu-112389DiVA: diva2:286094
Available from: 2010-01-13 Created: 2010-01-13 Last updated: 2010-01-19Bibliographically approved

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