n-Kirchhoff type equations with exponential nonlinearities

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• 作者：Sarika Goyal ; Pawan Kumar Mishra…
• 关键词：Kirchhoff equation ; Trudinger ; Moser embedding ; Sign ; changing weight function
• 刊名：Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：110
• 期：1
• 页码：219-245
• 全文大小：603 KB
• 参考文献：1.Adimurthi, A.: Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the \(n\) -Laplacian. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17, 393–413 (1990)MathSciNet MATH
  2.Adimurthi, A., Sandeep, K.: A singular Moser-Trudinger embedding and its applications. Nonlinear Differ. Equ. Appl. 13, 585–603 (2007)CrossRef MathSciNet MATH
  3.Alves, C.O., Corrêa, F.J.S.A., Figueiredo, G.M.: On a class of nonlocal elliptic problems with critical growth. DEA 2, 409–417 (2010)MATH
  4.Alves, C.O., Corrêa, F.J.S.A., Ma, T.F.: Positive solutions for a quasilinear elliptic equation of Kirchhoff type. Comput. Math. Appl. 49, 85–93 (2005)CrossRef MathSciNet MATH
  5.Alves, C.O., El Hamidi, A.: Nehari manifold and existence of positive solutions to a class of quasilinear problem. Nonlinear Anal. 60(4), 611–624 (2005)CrossRef MathSciNet MATH
  6.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)CrossRef MathSciNet MATH
  7.Brown, K.J., Zhang, Y.: The Nehari manifold for a semilinear elliptic problem with a sign-changing weight function. J. Differ. Equ. 193, 481–499 (2003)CrossRef MathSciNet MATH
  8.Brown, K.J., Wu, T.F.: A fibering map approach to a semilinear elliptic boundry value problem. Electron. J. Differ. Equ. 69, 1–9 (2007)MathSciNet
  9.Chen, C., Kuo, Y., Wu, T.: The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions. J. Differ. Equ. 250(4), 1876–1908 (2011)CrossRef MathSciNet MATH
  10.Cheng, B.T., Wu, X., Liu, J.: Multiple solutions for a class of Kirchhoff type problems with concave nonlinearity. Nonlinear Differ. Equ. Appl. 19(5), 521–537 (2012)CrossRef MathSciNet MATH
  11.Cheng, B.T., Wu, X.: Existence results of positive solutions of Kirchhoff problems. Nonlinear Anal. 71, 4883–4892 (2009)CrossRef MathSciNet MATH
  12.Chen, C.S., Huang, J.C., Liu, L.H.: Multiple solutions to the nonhomogeneous \(p\) -Kirchhoff elliptic equation with concave-convex nonlinearities. Appl. Math. Lett. 26(7), 754–759 (2013)CrossRef MathSciNet MATH
  13.Chipot, M., Lovat, B.: Some remarks on nonlocal elliptic and parabolic problems. Nonlinear Anal. 30(7), 4619–4627 (1997)CrossRef MathSciNet MATH
  14.Corrêa, F.J.S.A., Figueiredo, G.M.: On an elliptic equation of \(p\) -Kirchhoff-type via variational methods. Bull. Austral. Math. Soc. 77, 263–277 (2006)
  15.Corrêa, F.J.S.A.: On positive solutions of nonlocal and nonvariational elliptic problems. Nonlinear Anal. 59, 1147–1155 (2004)CrossRef MathSciNet MATH
  16.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)CrossRef MATH
  17.Drabek, P., Pohozaev, S.I.: Positive solutions for the p-Laplacian: application of the fibering method. Proc. Royal Soc. Edinburgh Sect A 127, 703–726 (1997)CrossRef MathSciNet MATH
  18.El Hamidi, A.: Multiple solutions with changing sign energy to a nonlinear elliptic equation. Commun. Pure Appl. Anal. 3, 253–265 (2004)CrossRef MathSciNet MATH
  19.Figueiredo, G.M.: Ground state soluttion for a Kirchhoff problem with exponential critical growth, arXiv:​1305.​2571v1 [math.AP]
  20.Giacomoni, J., Sreenadh, K.: A multiplicity result to a nonhomogeneous elliptic equation in whole space \({\mathbb{R}}^2\) . Adv. Math. Sci. Appl. 15(2), 467–488 (2005)MathSciNet MATH
  21.Giacomoni, J., Prashanth, S., Sreenadh, K.: A global multiplicity result for \(N\) -Laplacian with critical nonlinearity of concave-convex type. J. Differ. Equ. 232, 544–572 (2007)CrossRef MathSciNet MATH
  22.He, X., Zou, W.: Existence of a positive solution to Kirchhoff type problems without compactness conditions. J. Differ. Equ. 253(7), 2285–2294 (2012)CrossRef MathSciNet
  23.He, X., Zou, W.: Existence and concentration behavior of positive solutions for a Kirchhoff equation in \({\mathbb{R}}^3\) . J. Differ. Equ. 252(2), 1813–1834 (2012)CrossRef MathSciNet MATH
  24.Li, Y., Li, F., Shi, J.: Infinitely many positive solutions for Kirchhoff-type problems. Nonlinear Anal. 70, 1407–1414 (2009)CrossRef MathSciNet
  25.Lions, P.L.: The concentration compactness principle in the calculus of variations part-I Rev. Mat. Iberoamericana 1, 185–201 (1985)
  26.Marcos do Ó, J.: Semilinear Dirichlet problems for the \(N\) -Laplacian in \(\Omega \) with nonlinearities in critical growth range. Differ. Integral Equ. 9, 967–979 (1996)MATH
  27.Moser, J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20, 1077–1092 (1971)CrossRef
  28.Prashanth, S., Sreenadh, K.: Multiplicity of Solutions to a nonhomogeneous elliptic equation in \({\mathbb{R}}^2\) . Differ. Integral Equ. 18, 681–698 (2005)MathSciNet MATH
  29.Tarantello, G.: On nonhomogeneous elliptic equations involving critical Sobolev exponent. Ann. Inst. H. Poincare- Anal. non lineaire 9, 281–304 (1992)MathSciNet MATH
  30.Wu, T.F.: On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function. J. Math. Anal. Appl. 318, 253–270 (2006)CrossRef MathSciNet MATH
  31.Wu, T.F.: Multiplicity results for a semilinear elliptic equation involving sign-changing weight function. Rocky Mountain J. Math. 39(3), 995–1011 (2009)CrossRef MathSciNet MATH
  32.Wu, T.F.: Multiple positive solutions for a class of concave-convex elliptic problems in \(\Omega \) involving sign-changing weight. J. Funct. Anal. 258(1), 99–131 (2010)CrossRef MathSciNet MATH
  
• 作者单位：Sarika Goyal (1)
  Pawan Kumar Mishra (1)
  K. Sreenadh (1)
  
  1. Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, 110 016, India
  
• 刊物类别：Mathematics and Statistics
• 出版者：Springer Milan
• ISSN：1579-1505

文摘

In this article, we study the existence of non-negative solutions of the class of non-local problem of n-Kirchhoff type $$\begin{aligned} \left\{ \begin{array}{l} -m(\int _{\Omega }|\nabla u|^n)\Delta _n u = f(x,u) \; \text {in}\; \Omega ,\quad u =0\quad \text {on} \quad \partial \Omega , \end{array} \right. \end{aligned}$$where \(\Omega \subset \mathbb R^n\) is a bounded domain with smooth boundary, \(n\ge 2\) and f behaves like \(e^{|u|^{\frac{n}{n-1}}}\) as \(|u|\rightarrow \infty \). Moreover, by minimization on the suitable subset of the Nehari manifold, we study the existence and multiplicity of solutions, when f(x, t) is concave near \(t=0\) and convex as \(t\rightarrow \infty \). Keywords Kirchhoff equation Trudinger-Moser embedding Sign-changing weight function Mathematics Subject Classification 35J35 35J60 35J92 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 (32) References1.Adimurthi, A.: Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the \(n\)-Laplacian. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17, 393–413 (1990)MathSciNetMATH2.Adimurthi, A., Sandeep, K.: A singular Moser-Trudinger embedding and its applications. Nonlinear Differ. Equ. Appl. 13, 585–603 (2007)CrossRefMathSciNetMATH3.Alves, C.O., Corrêa, F.J.S.A., Figueiredo, G.M.: On a class of nonlocal elliptic problems with critical growth. DEA 2, 409–417 (2010)MATH4.Alves, C.O., Corrêa, F.J.S.A., Ma, T.F.: Positive solutions for a quasilinear elliptic equation of Kirchhoff type. Comput. Math. Appl. 49, 85–93 (2005)CrossRefMathSciNetMATH5.Alves, C.O., El Hamidi, A.: Nehari manifold and existence of positive solutions to a class of quasilinear problem. Nonlinear Anal. 60(4), 611–624 (2005)CrossRefMathSciNetMATH6.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)CrossRefMathSciNetMATH7.Brown, K.J., Zhang, Y.: The Nehari manifold for a semilinear elliptic problem with a sign-changing weight function. J. Differ. Equ. 193, 481–499 (2003)CrossRefMathSciNetMATH8.Brown, K.J., Wu, T.F.: A fibering map approach to a semilinear elliptic boundry value problem. Electron. J. Differ. Equ. 69, 1–9 (2007)MathSciNet9.Chen, C., Kuo, Y., Wu, T.: The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions. J. Differ. Equ. 250(4), 1876–1908 (2011)CrossRefMathSciNetMATH10.Cheng, B.T., Wu, X., Liu, J.: Multiple solutions for a class of Kirchhoff type problems with concave nonlinearity. Nonlinear Differ. Equ. Appl. 19(5), 521–537 (2012)CrossRefMathSciNetMATH11.Cheng, B.T., Wu, X.: Existence results of positive solutions of Kirchhoff problems. Nonlinear Anal. 71, 4883–4892 (2009)CrossRefMathSciNetMATH12.Chen, C.S., Huang, J.C., Liu, L.H.: Multiple solutions to the nonhomogeneous \(p\)-Kirchhoff elliptic equation with concave-convex nonlinearities. Appl. Math. Lett. 26(7), 754–759 (2013)CrossRefMathSciNetMATH13.Chipot, M., Lovat, B.: Some remarks on nonlocal elliptic and parabolic problems. Nonlinear Anal. 30(7), 4619–4627 (1997)CrossRefMathSciNetMATH14.Corrêa, F.J.S.A., Figueiredo, G.M.: On an elliptic equation of \(p\)-Kirchhoff-type via variational methods. Bull. Austral. Math. Soc. 77, 263–277 (2006)15.Corrêa, F.J.S.A.: On positive solutions of nonlocal and nonvariational elliptic problems. Nonlinear Anal. 59, 1147–1155 (2004)CrossRefMathSciNetMATH16.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)CrossRefMATH17.Drabek, P., Pohozaev, S.I.: Positive solutions for the p-Laplacian: application of the fibering method. Proc. Royal Soc. Edinburgh Sect A 127, 703–726 (1997)CrossRefMathSciNetMATH18.El Hamidi, A.: Multiple solutions with changing sign energy to a nonlinear elliptic equation. Commun. Pure Appl. Anal. 3, 253–265 (2004)CrossRefMathSciNetMATH19.Figueiredo, G.M.: Ground state soluttion for a Kirchhoff problem with exponential critical growth, arXiv:​1305.​2571v1[math.AP]20.Giacomoni, J., Sreenadh, K.: A multiplicity result to a nonhomogeneous elliptic equation in whole space \({\mathbb{R}}^2\). Adv. Math. Sci. Appl. 15(2), 467–488 (2005)MathSciNetMATH21.Giacomoni, J., Prashanth, S., Sreenadh, K.: A global multiplicity result for \(N\)-Laplacian with critical nonlinearity of concave-convex type. J. Differ. Equ. 232, 544–572 (2007)CrossRefMathSciNetMATH22.He, X., Zou, W.: Existence of a positive solution to Kirchhoff type problems without compactness conditions. J. Differ. Equ. 253(7), 2285–2294 (2012)CrossRefMathSciNet23.He, X., Zou, W.: Existence and concentration behavior of positive solutions for a Kirchhoff equation in \({\mathbb{R}}^3\). J. Differ. Equ. 252(2), 1813–1834 (2012)CrossRefMathSciNetMATH24.Li, Y., Li, F., Shi, J.: Infinitely many positive solutions for Kirchhoff-type problems. Nonlinear Anal. 70, 1407–1414 (2009)CrossRefMathSciNet25.Lions, P.L.: The concentration compactness principle in the calculus of variations part-I Rev. Mat. Iberoamericana 1, 185–201 (1985)26.Marcos do Ó, J.: Semilinear Dirichlet problems for the \(N\)-Laplacian in \(\Omega \) with nonlinearities in critical growth range. Differ. Integral Equ. 9, 967–979 (1996)MATH27.Moser, J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20, 1077–1092 (1971)CrossRef28.Prashanth, S., Sreenadh, K.: Multiplicity of Solutions to a nonhomogeneous elliptic equation in \({\mathbb{R}}^2\). Differ. Integral Equ. 18, 681–698 (2005)MathSciNetMATH29.Tarantello, G.: On nonhomogeneous elliptic equations involving critical Sobolev exponent. Ann. Inst. H. Poincare- Anal. non lineaire 9, 281–304 (1992)MathSciNetMATH30.Wu, T.F.: On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function. J. Math. Anal. Appl. 318, 253–270 (2006)CrossRefMathSciNetMATH31.Wu, T.F.: Multiplicity results for a semilinear elliptic equation involving sign-changing weight function. Rocky Mountain J. Math. 39(3), 995–1011 (2009)CrossRefMathSciNetMATH32.Wu, T.F.: Multiple positive solutions for a class of concave-convex elliptic problems in \(\Omega \) involving sign-changing weight. J. Funct. Anal. 258(1), 99–131 (2010)CrossRefMathSciNetMATH About this Article Title n-Kirchhoff type equations with exponential nonlinearities Journal Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Volume 110, Issue 1 , pp 219-245 Cover Date2016-03 DOI 10.1007/s13398-015-0230-x Print ISSN 1578-7303 Online ISSN 1579-1505 Publisher Springer Milan Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Mathematics, general Applications of Mathematics Theoretical, Mathematical and Computational Physics Keywords Kirchhoff equation Trudinger-Moser embedding Sign-changing weight function 35J35 35J60 35J92 Authors Sarika Goyal (1) Pawan Kumar Mishra (1) K. Sreenadh (1) Author Affiliations 1. Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, 110 016, India Continue reading... To view the rest of this content please follow the download PDF link above.