Multiple positive solutions of singular and critical elliptic problem in {mathbb{R}^2} with discontinuous nonlinearities

摘要

Let Ω be a bounded domain in ({mathbb{R}^2}) with smooth boundary. We consider the following singular and critical elliptic problem with discontinuous nonlinearity:$$(P_lambda)left {begin{array}{ll} - Delta u = lambda left(frac{m(x, u) e^{alpha{u}^2}}{|x|^{beta}} + u^{q}g(u - a)right),quad{u} > 0 quad {rm in} quad Omegau quad quad = 0quad {rm on} quad partial Omega end{array}right.$$where ({0leq q < 1 ,0< alphaleq4pi}) and ({beta in [0, 2)}) such that ({frac{beta}{2} + frac{alpha}{4pi} leq 1}) and ({{g(t - a) = left{begin{array}{ll}1, t leq a 0, t > a.end{array}right.}}) Under the suitable assumptions on m(x, t) we show the existence and multiplicity of solutions for maximal interval for λ. Mathematics Subject Classification (2000) 35J20 35J65 Keywords Singular elliptic problem Critical growth Discontinuous nonlinearity 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 (17) References1.Adimurthi: Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the n-Laplacian. Annali della Scuola Sup. di Pisa Serie IV, Vol. 17, no. 3, pp. 393–413 (1990)2.Adimurthi, Giacomoni J.: Multiplicity of positive solutions for a singular and critical elliptic problem in ({{mathbb{R}^2}}). Commun. Contemp. Math. 8(5), 621–656 (2006)3.Adimurthi, Sandeep, K.: A singular Moser–Trudinger embedding and its applications, NoDEA 13, pp 585–603 (2007)4.Ambrosetti A., Brezis H., Cerami G.: Combined effects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal. 122(2), 519–543 (1994)MathSciNetCrossRefMATH5.Badiale M., Tarantello G.: Existence and multiplicity results for elliptic problems with critical growth and discontinuous nonlinearities. Nonlinear Anal. 29(6), 639–677 (1997)MathSciNetCrossRefMATH6.Sumit Kaur, B.; Sreenadh, K.: Multiple positive solutions for a singular elliptic equation with Neumann boundary condition in two dimensions. Electron. J. Differ. Equ. 2009(43), 1–13 (2009)7.Brezis H., Nirenberg L.: H 1 versus C 1 local minimizers. C. R. Acad. Sci. Paris Sr. I Math. 317(5), 465–472 (1993)MathSciNetMATH8.Chang K.C.: Variational methods for nondifferentiable functionals and their applications to partial differential equations. J. Math. Anal. Appl. 80(1), 102–129 (1981)MathSciNetCrossRefMATH9.Clarke F.H.: Generalized gradients and applications. Trans. Am. Math. Soc. 205, 247–262 (1975)CrossRefMATH10.de Figueiredo D.G., Miyagaki O.H., Ruf B.: Elliptic equations in ({mathbb{R}^2}) with nonlinearities in the critical growth range. Calc. Var. Partial Differ. Equ. 3(2), 139–153 (1995)CrossRefMATH11.Ghoussoub N., Preiss D.: A general mountain pass principle for locating and classifying critical points. Ann. Inst. H. Poincaré Anal. Non Linéaire 6(5), 321–330 (1989)MathSciNetMATH12.Giacomoni J., Prashanth S., Sreenadh K.: A global multiplicity result for N-Laplacian with critical nonlinearity of concave-convex type. J. Differ. Equ. 232(2), 544–572 (2007)MathSciNetCrossRefMATH13.Haitao Y.: Multiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic problem. J. Differ. Equ. 189(2), 487–512 (2003)MathSciNetCrossRefMATH14.Haitao Y., Shaoping W.: An elliptic problem with discontinuous sublinear nonlinearities in ({mathbb{R}^N}). Nonlinear Anal. 51(6), 921–939 (2002)MathSciNetCrossRefMATH15.Moser J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20, 1077–1092 (1971)CrossRef16.Sandeep K.: On the first eigenfunction of a perturbed Hardy–Sobolev operator. NoDEA 10(2), 223–253 (2003)MathSciNetMATH17.Trudinger N.S.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17, 473–483 (1967)MathSciNetMATH About this Article Title Multiple positive solutions of singular and critical elliptic problem in ({mathbb{R}^2}) with discontinuous nonlinearities Journal Nonlinear Differential Equations and Applications NoDEA Volume 20, Issue 6 , pp 1831-1850 Cover Date2013-12 DOI 10.1007/s00030-013-0232-3 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 35J20 35J65 Singular elliptic problem Critical growth Discontinuous nonlinearity Authors K. Sreenadh (1) Sweta Tiwari (1) Author Affiliations 1. Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khaz, New Delhi, 110016, India Continue reading... To view the rest of this content please follow the download PDF link above.