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Quasilinear elliptic equations and weighted Sobolev-Poincare inequalities with distributional weights
Kent State University, OH 44240 USA .
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
University of Missouri, MO 65211 USA .
2013 (English)In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 232, no 1, 513-542 p.Article in journal (Refereed) Published
Abstract [en]

We introduce a class of weak solutions to the quasilinear equation -Delta(p)u = sigma vertical bar u vertical bar(P-2)u in an open set Omega subset of R-n with p andgt; 1, where Delta(p)u = del. (vertical bar del u vertical bar(p-2)del u) is the p-Laplacian operator. Our notion of solution is tailored to general distributional coefficients sigma which satisfy the inequality less thanbrgreater than less thanbrgreater than-Lambda integral(Omega) vertical bar del h vertical bar(p)dx andlt;= andlt;vertical bar h vertical bar(p), sigma andgt; andlt;= lambda integral(Omega) vertical bar del h vertical bar(p)dx, less thanbrgreater than less thanbrgreater thanfor all h is an element of C-0(infinity)(Omega). Here 0 andlt; Lambda andlt; +infinity-, and less thanbrgreater than less thanbrgreater than0 andlt; lambda andlt; (p - 1)(2-p) if p andgt;= 2, or 0 andlt; lambda andlt; 1 if 0 andlt; p andlt; 2. less thanbrgreater than less thanbrgreater thanAs we shall demonstrate, these conditions on lambda are natural for the existence of positive solutions, and cannot be relaxed in general. Furthermore, our class of solutions possesses the optimal local Sobolev regularity available under such a mild restriction on sigma. less thanbrgreater than less thanbrgreater thanWe also study weak solutions of the closely related equation -Delta p nu = (p - 1)vertical bar del nu vertical bar(p) +sigma, under the same conditions on . Our results for this latter equation will allow us to characterize the class of sigma satisfying the above inequality for positive lambda and Lambda. thereby extending earlier results on the form boundedness problem for the Schrodinger operator to p not equal 2.

Place, publisher, year, edition, pages
Elsevier , 2013. Vol. 232, no 1, 513-542 p.
Keyword [en]
Quasilinear equations, Weighted integral inequalities, Elliptic regularity
National Category
Natural Sciences
URN: urn:nbn:se:liu:diva-86621DOI: 10.1016/j.aim.2012.09.029ISI: 000311268300018OAI: diva2:580122

Funding Agencies|NSF|DMS-0901550|

Available from: 2012-12-20 Created: 2012-12-20 Last updated: 2012-12-20

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Maz´ya, Vladimir
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