Existence Results for a Class of p(x)- Kirchhoff Problem with a Singular Weight

详细信息    查看全文

• 作者：Mostafa Allaoui ; Anass Ourraoui
• 关键词：p(x) ; Kirchhoff ; fountain theorem
• 刊名：Mediterranean Journal of Mathematics
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：13
• 期：2
• 页码：677-686
• 全文大小：518 KB
• 参考文献：1.Afrouzi, G.A., Mirzapour, M.: Eigenvalue problems for p(x)-Kirchhoff type equations. Electron. J. Differ. Equ. 2013(253), 1–10 (2013)
  2.Allaoui, M., El Amrouss, A., Ourraoui, A.: On a class of nonlocal p(x)- Laplacian Neumann problems. Thai J. Math. (2015, in press)
  3.Autuori, G., Colasuonno, F., Pucci, P.: On the existence of stationary solutions for higher-order p-Kirchhoff problems. Commun. Contemp. Math. 16(5), 1–43 (2014). Art. Id 1450002
  4.Autuori G., Pucci P., Salvatori M.C.: Asymptotic stability for anisotropic Kirchhoff systems. J. Math. Anal. Appl. 352, 149–165 (2009)MathSciNet CrossRef MATH
  5.Cammaroto F., Vilasi L.: Multiple solutions for a Kirchhoff-type problem involving the p(x)-Laplacian operator. Nonlinear Anal. 74, 1841–1852 (2011)MathSciNet CrossRef MATH
  6.Chen Y., Levine S., Rao R.: Variable exponent, Linear growth functionals in image processing. SIAM J. Appl. Math. 66, 1383–1406 (2006)MathSciNet CrossRef MATH
  7.Colasuonno F., Pucci P.: Multiplicity of solutions for p(x)-polyharmonic elliptic Kirchhoff equations. Nonlinear Anal. 74, 5962–5974 (2011)MathSciNet CrossRef MATH
  8.Dai G., Ma R.: Solutions for a p(x)-Kirchhoff type equation with Neumann boundary data. Nonlinear Anal. Real World Appl. 12, 2666–2680 (2011)MathSciNet CrossRef MATH
  9.Diening, L., Harjulehto, P., Hasto, P., Ruzicka, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics, vol. 2017. Springer, Berlin (2011)
  10.Fan X.: On nonlocal p(x)-Laplacian Dirichlet problems. Nonlinear Anal. 72, 3314–3323 (2010)MathSciNet CrossRef MATH
  11.Fan X.: Solutions for p(x)-Laplacian Dirichlet problems with singular coefficients. J. Math. Anal. Appl. 312, 464–477 (2005)MathSciNet CrossRef MATH
  12.Fan X., Shen J., Zhao D.: Sobolev embedding theorems for spaces \({W^{k,p(x)}(\Omega)}\) . J. Math. Anal. Appl. 262, 749–760 (2001)MathSciNet CrossRef MATH
  13.Huang J., Chen C., Xiu Z.: Existence and multiplicity results for a p-Kirchhoff equation with a concave–convex term. Appl. Math. Lett. 26, 1070–1075 (2013)MathSciNet CrossRef MATH
  14.Kirchhoff, G.: Mechanik. Teubner, Leipzig (1883)
  15.Kov\({\breve{\check{{\rm a} }}\check{{\rm c}}}\) ik, O., R\({\breve{{\rm a}}}\) kosnk, J.: On spaces \({L^{p(x)}(\Omega)}\) and \({W^{k,p(x)}(\Omega)}\) . Czechoslov. Math. J. 41, 592–618 (1991)
  16.Mashiyev, R.A., Cekic, B., Buhrii, O.M.: Existence of solutions for p(x)- Laplacian equations. Electron. J. Qual. Theory Differ. Equ. (65), 1–13 (2010)
  17.Ružička, M.: Flow of shear dependent electrorheological fluids. C. R. Math. Acad. Sci. Paris 329, 393–398 (1999)
  18.Willem, M.: Minimax Theorems. Birkhäuser, Boston (1996)
  19.Zhikov V.: Averaging of functionals in the calculus of variations and elasticity. Math. USSR Izv. 29, 33–66 (1987)CrossRef MATH
  
• 作者单位：Mostafa Allaoui (1)
  Anass Ourraoui (1)
  
  1. Department of Mathematics and Computer Science, Faculty of Sciences, University Mohamed I, Oujda, Morocco
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Birkh盲user Basel
• ISSN：1660-5454

文摘

The present article is concerned with the following nonlocal elliptic equation involving concave and convex terms, $$\begin{array}{ll}- M \left(\int_\Omega \frac{1}{p(x)}|\nabla u|^{p(x)}{\rm d}x\right)\Big(\Delta_{p(x)}u\Big) \!&=\! \lambda \big(g(x)|u|^{q(x)-2}u\!-\!h(x)\\ &\quad |u|^{r(x)-2}u\big), \quad x\in \Omega,\\ & u = 0,\quad x\in \partial\Omega. \end{array}$$By means of the variational approach, we prove that the above problem admits a sequence of infinitely many solutions under suitable assumptions. Keywords p(x)-Kirchhoff fountain theorem Mathematics Subject Classification 35J60 35D05 35J20 35J40 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 (19) References1.Afrouzi, G.A., Mirzapour, M.: Eigenvalue problems for p(x)-Kirchhoff type equations. Electron. J. Differ. Equ. 2013(253), 1–10 (2013)2.Allaoui, M., El Amrouss, A., Ourraoui, A.: On a class of nonlocal p(x)- Laplacian Neumann problems. Thai J. Math. (2015, in press)3.Autuori, G., Colasuonno, F., Pucci, P.: On the existence of stationary solutions for higher-order p-Kirchhoff problems. Commun. Contemp. Math. 16(5), 1–43 (2014). Art. Id 14500024.Autuori G., Pucci P., Salvatori M.C.: Asymptotic stability for anisotropic Kirchhoff systems. J. Math. Anal. Appl. 352, 149–165 (2009)MathSciNetCrossRefMATH5.Cammaroto F., Vilasi L.: Multiple solutions for a Kirchhoff-type problem involving the p(x)-Laplacian operator. Nonlinear Anal. 74, 1841–1852 (2011)MathSciNetCrossRefMATH6.Chen Y., Levine S., Rao R.: Variable exponent, Linear growth functionals in image processing. SIAM J. Appl. Math. 66, 1383–1406 (2006)MathSciNetCrossRefMATH7.Colasuonno F., Pucci P.: Multiplicity of solutions for p(x)-polyharmonic elliptic Kirchhoff equations. Nonlinear Anal. 74, 5962–5974 (2011)MathSciNetCrossRefMATH8.Dai G., Ma R.: Solutions for a p(x)-Kirchhoff type equation with Neumann boundary data. Nonlinear Anal. Real World Appl. 12, 2666–2680 (2011)MathSciNetCrossRefMATH9.Diening, L., Harjulehto, P., Hasto, P., Ruzicka, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics, vol. 2017. Springer, Berlin (2011)10.Fan X.: On nonlocal p(x)-Laplacian Dirichlet problems. Nonlinear Anal. 72, 3314–3323 (2010)MathSciNetCrossRefMATH11.Fan X.: Solutions for p(x)-Laplacian Dirichlet problems with singular coefficients. J. Math. Anal. Appl. 312, 464–477 (2005)MathSciNetCrossRefMATH12.Fan X., Shen J., Zhao D.: Sobolev embedding theorems for spaces \({W^{k,p(x)}(\Omega)}\) . J. Math. Anal. Appl. 262, 749–760 (2001)MathSciNetCrossRefMATH13.Huang J., Chen C., Xiu Z.: Existence and multiplicity results for a p-Kirchhoff equation with a concave–convex term. Appl. Math. Lett. 26, 1070–1075 (2013)MathSciNetCrossRefMATH14.Kirchhoff, G.: Mechanik. Teubner, Leipzig (1883)15.Kov\({\breve{\check{{\rm a} }}\check{{\rm c}}}\)ik, O., R\({\breve{{\rm a}}}\)kosnk, J.: On spaces \({L^{p(x)}(\Omega)}\) and \({W^{k,p(x)}(\Omega)}\) . Czechoslov. Math. J. 41, 592–618 (1991)16.Mashiyev, R.A., Cekic, B., Buhrii, O.M.: Existence of solutions for p(x)- Laplacian equations. Electron. J. Qual. Theory Differ. Equ. (65), 1–13 (2010)17.Ružička, M.: Flow of shear dependent electrorheological fluids. C. R. Math. Acad. Sci. Paris 329, 393–398 (1999)18.Willem, M.: Minimax Theorems. Birkhäuser, Boston (1996)19.Zhikov V.: Averaging of functionals in the calculus of variations and elasticity. Math. USSR Izv. 29, 33–66 (1987)CrossRefMATH About this Article Title Existence Results for a Class of p(x)- Kirchhoff Problem with a Singular Weight Journal Mediterranean Journal of Mathematics Volume 13, Issue 2 , pp 677-686 Cover Date2016-04 DOI 10.1007/s00009-015-0518-2 Print ISSN 1660-5446 Online ISSN 1660-5454 Publisher Springer International Publishing Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Mathematics, general Keywords 35J60 35D05 35J20 35J40 p(x)-Kirchhoff fountain theorem Authors Mostafa Allaoui (1) Anass Ourraoui (1) Author Affiliations 1. Department of Mathematics and Computer Science, Faculty of Sciences, University Mohamed I, Oujda, Morocco Continue reading... To view the rest of this content please follow the download PDF link above.