{W^{1,1}_0} -solutions for elliptic problems having gradient quadratic lower order terms

摘要

In this paper we deal with solutions of problems of the type $$left{begin{array}{ll}-{rm div} Big(frac{a(x)Du}{(1+|u|)^2} Big)+u = frac{b(x)|Du|^2}{(1+|u|)^3} +f quad &{rm in} , Omega, u=0 &{rm on} partial , Omega, end{array} right.$$where ({0 < alpha leq a(x) leq beta, |b(x)| leq gamma, gamma > 0, f in L^2 (Omega)}) and Ω is a bounded subset of ({mathbb{R}^N}) with N ≥ 3. We prove the existence of at least one solution for such a problem in the space ({W_{0}^{1, 1}(Omega) cap L^{2}(Omega)}) if the size of the lower order term satisfies a smallness condition when compared with the principal part of the operator. This kind of problems naturally appears when one looks for positive minima of a functional whose model is: $$J (v) = frac{alpha}{2} int_{Omega}frac{|D v|^2}{(1 + |v|)^{2}} + frac{12}{int_{Omega}|v|^2} - int_{Omega}f,v , quad f in L^2(Omega),$$where in this case a(x) ≡ b(x) = α > 0. Mathematics Subject Classification 35D30 35J65 35J70 Keywords Nonlinear elliptic equations ({W_{0}^{1, 1} (Omega)}) solutions Quadratic gradient terms Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (16) References1.Arcoya D., Boccardo L., Leonori T., Porretta A.: Some elliptic problems with singular and gradient quadratic lower order terms. J. Diff. Equ. 249, 2771–2795 (2010)MathSciNetCrossRefMATH2.Boccardo, L.: Some nonlinear Dirichlet problems in L 1 involving lower order terms in divergence form. Progress in elliptic and parabolic partial differential equations (Capri, 1994), pp. 43–57. 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Fourier (Grenoble) 15(1), 189–258 (1965)MathSciNetCrossRefMATH About this Article Title ({W^{1,1}_0}) -solutions for elliptic problems having gradient quadratic lower order terms Journal Nonlinear Differential Equations and Applications NoDEA Volume 20, Issue 6 , pp 1741-1757 Cover Date2013-12 DOI 10.1007/s00030-013-0228-z Print ISSN 1021-9722 Online ISSN 1420-9004 Publisher Springer Basel Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Analysis Keywords 35D30 35J65 35J70 Nonlinear elliptic equations $${W_{0}^{1, 1} (Omega)}$$ solutions Quadratic gradient terms Authors David Arcoya (1) Lucio Boccardo (2) Tommaso Leonori (3) Author Affiliations 1. Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, Campus Fuentenueva S/N, 18071, Granada, Spain 2. Dipartimento di Matematica, Università di Roma “La Sapienza”, Piazza A. Moro 2, 00185, Rome, Italy 3. Departamento di Matematicas, Universidad Carlos III, Avda. de la Universidad 30, 28911, Leganés Madrid, Spain Continue reading... To view the rest of this content please follow the download PDF link above.