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Multiplicity of solutions to elliptic equations with exponential nonlinearities

Funds:  The Natural Science Foundation of NNSF funds of NNSF of China (11526177), the Natural Science Foundation of Jiangsu Province (BK20151160), the Nurturing Project of Xuzhou University of Technology (XKY2019103).
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https://doi.org/10.3969/j.issn.0253-2778.2020.03.007
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  • Corresponding author: XUE Yimin(corresponding author), male, born in 1977, master/lecturer. Research field: Differential equations and their applications. E-mail: xueym@xzit.edu.cn
  • Received Date: 23 March 2019
  • Accepted Date: 14 June 2019
  • Rev Recd Date: 14 June 2019
  • Publish Date: 31 March 2020
    Abstract

  • Abstract

    The multiplicity of positive solutions to quasi-linear elliptic equations with exponential nonlinearities is obtained through a singular Trudinger-Moser inequality, which is due to Ref.[21], the mountain-pass theorem without the Palais-Smale condition and the Ekeland’s variational principle. In particular, for the proof of our main results, we follow the lines of Refs.[11,15].

    Abstract

    The multiplicity of positive solutions to quasi-linear elliptic equations with exponential nonlinearities is obtained through a singular Trudinger-Moser inequality, which is due to Ref.[21], the mountain-pass theorem without the Palais-Smale condition and the Ekeland’s variational principle. In particular, for the proof of our main results, we follow the lines of Refs.[11,15].
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    • References(25)
  • [1]
    MOSER J. A sharp form of an inequality by N.Trudinger[J]. Indiana University Mathematics Journal, 1971, 20(11): 1077-1091.
    [2]
    PEETRE J. Espaces d’interpolation et theoreme de Soboleff[J]. Annales Inst Fourier(Grenoble), 1966, 16(1): 279-317.
    [3]
    POHOZAEV S. The Sobolev embedding in the special case pl = n[C]// Proceedings of the Technical Scientific Conference on Advances of Scientific Reseach. Moscow: Energet. Inst., 1965: 158-170.
    [4]
    TRUDINGER N. On embeddings into Orlicz spaces and some applications[J]. Journal of Mathematical Fluid Mechanics, 1967, 17: 473-484.
    [5]
    YUDOVICH V. Some estimates connected with integral operators and with solutions of elliptic equations[J]. Doklady Akademii Nauk SSSR, 1961, 138(4): 805-808.
    [6]
    ADIMURTHI, SANDEEP K. A singular Moser-Trudinger embedding and its applications[J]. Non-linear Differential Equations and Applications NoDEA, 2007, 13(5): 585-603.
    [7]
    PANDA R. Nontrivial solution of a quasilinear elliptic equation with critical growth in RN[J]. Proceedings of the Indian Academy of Sciences(MathematicalSciences), 1995, 105(4): 425-444.
    [8]
    DO ?倕O J M. N-Laplacian equations in RN with critical growth[J]. Abstract and Applied Analysis, 1997, 2(3-4): 301-315.
    [9]
    RUF B. A sharp Trudinger-Moser type inequality for unbounded domains in R2[J]. Journal of Functional Analysis, 2005, 219: 340-367.
    [10]
    LI Y, RUF B. A sharp Trudinger-Moser type inequality for unbounded domains in RN[J]. Indiana University Mathematics Journal, 2008, 57(1): 451-480.
    [11]
    ADIMURTHI, YANG Y. An interpolation of Hardy inequality and Trudinger-Moser inequality in RN and its applications[J]. International Mathematics Research Notices, 2010, 2010(13): 2394-2426.
    [12]
    SOUZA M DE, DO ?倕O J M. On a singular and nonhomogeneous N-Laplacian equation involving critical growth[J]. Journal of Mathematical Analysis and Applications, 2011, 380(1): 241-263.
    [13]
    DO ?倕O J M, SOUZA M DE. On a class of singular Trudinger-Moser type inequalities and its applications[J]. Mathematische Nachrichten, 2011, 284(14-15): 1754-1776.
    [14]
    DO ?倕O J M, MEDEIROS E, SEVERO U. On a quasilinear nonhomogeneous elliptic equation with critical growth in RN[J]. J Differential Equations, 2009, 246(4): 1363-1386.
    [15]
    YANG Y. Existence of positive solutions to quasi-linear elliptic equations with exponential growth in the whole Euclidean space[J]. Journal of Functional Analysis, 2012, 262(4): 1679-1704.
    [16]
    YANG Y. Trudinger-Moser inequalities on complete noncompact Riemannian manifolds[J]. Journal of Functional Analysis, 2012, 263(7): 1894-1938.
    [17]
    YANG Y. Adams type inequalities and related elliptic partial differential equations in dimension four[J].Journal of Differential Equations, 2012, 252(3): 2266-2295.
    [18]
    ZHAO L, CHANG Y. Min-max level estimate for a singular quasilinear polyharmonic equation in R2m[J]. Journal of Differential Equations, 2013, 254(6): 2434-2464.
    [19]
    SOUZAMDE. Some sharp inequalities related to Trudinger-Moser inequality[J]. Journal of Mathematical Analysis and Applications, 2018, 467: 981-1012.
    [20]
    LI X, YANG Y. Extremal functions for singular Trudinger-Moser inequalities in the entire Euclidean space[J]. Journal of Differential Equations,2018, 264: 4901-4943.
    [21]
    LI X. An improved singular Trudinger-Moser inequality in RN and its extremal functions[J]. Journal of Mathematical Analysis and Applications, 2018, 462: 1109-1129.
    [22]
    BREZIS H, LIEB E. A relation between pointwise convergence of functions and convergence of functionals[J]. Proceedings of the American Mathematical Society, 1983, 88(3): 486-490.
    [23]
    RABINOWITZ P H. On a class of nonlinear Schr¨odinger equations[J] Journal of Applied Mathematics and Physics(Zeitschrift f¨ur Angewandte Mathematik und Physik) ,1992, 43: 270-291.
    [24]
    WILLEM M. Minimax Theorems[M]. Boston: Birkhuser, 1996.
    [25]
    LINDQVIST P. Notes on the p-Laplace equation[R]. University of Jyvskyl Department of Mathematics and Statistics, 2006.)
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    [1]
    MOSER J. A sharp form of an inequality by N.Trudinger[J]. Indiana University Mathematics Journal, 1971, 20(11): 1077-1091.
    [2]
    PEETRE J. Espaces d’interpolation et theoreme de Soboleff[J]. Annales Inst Fourier(Grenoble), 1966, 16(1): 279-317.
    [3]
    POHOZAEV S. The Sobolev embedding in the special case pl = n[C]// Proceedings of the Technical Scientific Conference on Advances of Scientific Reseach. Moscow: Energet. Inst., 1965: 158-170.
    [4]
    TRUDINGER N. On embeddings into Orlicz spaces and some applications[J]. Journal of Mathematical Fluid Mechanics, 1967, 17: 473-484.
    [5]
    YUDOVICH V. Some estimates connected with integral operators and with solutions of elliptic equations[J]. Doklady Akademii Nauk SSSR, 1961, 138(4): 805-808.
    [6]
    ADIMURTHI, SANDEEP K. A singular Moser-Trudinger embedding and its applications[J]. Non-linear Differential Equations and Applications NoDEA, 2007, 13(5): 585-603.
    [7]
    PANDA R. Nontrivial solution of a quasilinear elliptic equation with critical growth in RN[J]. Proceedings of the Indian Academy of Sciences(MathematicalSciences), 1995, 105(4): 425-444.
    [8]
    DO ?倕O J M. N-Laplacian equations in RN with critical growth[J]. Abstract and Applied Analysis, 1997, 2(3-4): 301-315.
    [9]
    RUF B. A sharp Trudinger-Moser type inequality for unbounded domains in R2[J]. Journal of Functional Analysis, 2005, 219: 340-367.
    [10]
    LI Y, RUF B. A sharp Trudinger-Moser type inequality for unbounded domains in RN[J]. Indiana University Mathematics Journal, 2008, 57(1): 451-480.
    [11]
    ADIMURTHI, YANG Y. An interpolation of Hardy inequality and Trudinger-Moser inequality in RN and its applications[J]. International Mathematics Research Notices, 2010, 2010(13): 2394-2426.
    [12]
    SOUZA M DE, DO ?倕O J M. On a singular and nonhomogeneous N-Laplacian equation involving critical growth[J]. Journal of Mathematical Analysis and Applications, 2011, 380(1): 241-263.
    [13]
    DO ?倕O J M, SOUZA M DE. On a class of singular Trudinger-Moser type inequalities and its applications[J]. Mathematische Nachrichten, 2011, 284(14-15): 1754-1776.
    [14]
    DO ?倕O J M, MEDEIROS E, SEVERO U. On a quasilinear nonhomogeneous elliptic equation with critical growth in RN[J]. J Differential Equations, 2009, 246(4): 1363-1386.
    [15]
    YANG Y. Existence of positive solutions to quasi-linear elliptic equations with exponential growth in the whole Euclidean space[J]. Journal of Functional Analysis, 2012, 262(4): 1679-1704.
    [16]
    YANG Y. Trudinger-Moser inequalities on complete noncompact Riemannian manifolds[J]. Journal of Functional Analysis, 2012, 263(7): 1894-1938.
    [17]
    YANG Y. Adams type inequalities and related elliptic partial differential equations in dimension four[J].Journal of Differential Equations, 2012, 252(3): 2266-2295.
    [18]
    ZHAO L, CHANG Y. Min-max level estimate for a singular quasilinear polyharmonic equation in R2m[J]. Journal of Differential Equations, 2013, 254(6): 2434-2464.
    [19]
    SOUZAMDE. Some sharp inequalities related to Trudinger-Moser inequality[J]. Journal of Mathematical Analysis and Applications, 2018, 467: 981-1012.
    [20]
    LI X, YANG Y. Extremal functions for singular Trudinger-Moser inequalities in the entire Euclidean space[J]. Journal of Differential Equations,2018, 264: 4901-4943.
    [21]
    LI X. An improved singular Trudinger-Moser inequality in RN and its extremal functions[J]. Journal of Mathematical Analysis and Applications, 2018, 462: 1109-1129.
    [22]
    BREZIS H, LIEB E. A relation between pointwise convergence of functions and convergence of functionals[J]. Proceedings of the American Mathematical Society, 1983, 88(3): 486-490.
    [23]
    RABINOWITZ P H. On a class of nonlinear Schr¨odinger equations[J] Journal of Applied Mathematics and Physics(Zeitschrift f¨ur Angewandte Mathematik und Physik) ,1992, 43: 270-291.
    [24]
    WILLEM M. Minimax Theorems[M]. Boston: Birkhuser, 1996.
    [25]
    LINDQVIST P. Notes on the p-Laplace equation[R]. University of Jyvskyl Department of Mathematics and Statistics, 2006.)
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