March  2017, 16(2): 373-392. doi: 10.3934/cpaa.2017019

Singular periodic solutions for the p-laplacian ina punctured domain

1. 

School of Mathematics, South China University of Technology, Guangzhou 510640, China

2. 

Department of Mathematics, and Institute of Computational and Theoretical Studies, Hong Kong Baptist University, Kowloon, Hong Kong

3. 

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

* Corresponding author: RUI HUANG.

Received  June 2015 Revised  February 2016 Published  January 2017

Abstract. In this paper we are interested in studying singular periodic solutions for the p-Laplacian in a punctured domain. We find an interesting phenomenon that there exists a critical exponent pc = N and a singular exponent qs = p-1. Precisely speaking, only if p > pc can singular periodic solutions exist; while if 1 < ppc then all of the solutions have no singularity. By the singular exponent qs = p-1, we mean that in the case when q = qs, completely different from the remaining case qqs, the problem may or may not have solutions depending on the coefficients of the equation.

Citation: Shanming Ji, Yutian Li, Rui Huang, Xuejing Yin. Singular periodic solutions for the p-laplacian ina punctured domain. Communications on Pure & Applied Analysis, 2017, 16 (2) : 373-392. doi: 10.3934/cpaa.2017019
References:
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A. Beltramo and P. Hess, On the principal eigenvalue of a periodic-parabolic operator, Comm. Partial Differential Equations, 9 (1984), 919-941. doi: 10.1080/03605308408820351.  Google Scholar

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H. Brezis and L. Veron, Removable singularities for some nonlinear elliptic equations, Arch. Ration. Mech. Anal., 75 (1980/1981), 1-6. doi: 10.1007/BF00284616.  Google Scholar

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M. J. Esteban, On periodic solutions of superlinear parabolic problems, Trans. Amer. Math. Soc., 293 (1986), 171-189. doi: 10.2307/2000278.  Google Scholar

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M. J. Esteban, A remark on the existence of positive periodic solutions of superlinear parabolic problems, Proc. Amer. Math. Soc., 102 (1988), 131-136. doi: 10.2307/2046045.  Google Scholar

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B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  Google Scholar

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P. Lindqvist, On the equation div(|∇u|p-2u) + λ|u|p-2u = 0, Proc. Amer. Math. Soc., 109 (1990), 157-164. doi: 10.2307/2048375.  Google Scholar

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N. Mizoguchi, Periodic solutions for degenerate diffusion equations, Indiana Univ. Math. J., 44 (1995), 413-432. doi: 10.1512/iumj.1995.44.1994.  Google Scholar

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M. H. Protter and H. F. Weinberger, Maximum Principles in Differencial Equations, Prentice Hall, Englewood Cliffs, 1967. (Chinese Trans. , Science Press, Beijing, 1985). Google Scholar

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P. Quittner, Multiple equilibria, periodic solutions and a priori bounds for solutions in superlinear parabolic problems, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 237-258. doi: 10.1007/s00030-003-1056-3.  Google Scholar

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T. I. Seidman, Periodic solutions of a non-linear parabolic equation, J. Differential Equations, 19 (1975), 242-257.  Google Scholar

[14]

J. Serrin, Local behaior of solutions of quasilinear equations, Acta Math., 111 (1964), 247-302.  Google Scholar

[15]

J. Serrin, Isolated singularities of solutions of quasilinear equations, Acta Math., 113 (1965), 219-240.  Google Scholar

[16]

J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math., 189 (2002), 79-142. doi: 10.1007/BF02392645.  Google Scholar

[17]

J. L. Vàzquez and L. Véron, Removable singularities of some strongly nonlinear elliptic equations, Manuscripta Math., 33 (1980), 129-144. doi: 10.1007/BF01316972.  Google Scholar

[18]

N. Mizoguchi, Periodic solutions for degenerate diffusion equations, Indiana Univ. Math. J., 44 (1995), 413-432. doi: 10.1512/iumj.1995.44.1994.  Google Scholar

[19]

Y. F. Wang, J. X. Yin and Z. Q. Wu, Periodic solutions of evolution p-Laplacian equations with nonlinear sources, J. Math. Anal. Appl., 219 (1998), 76-96. doi: 10.1006/jmaa.1997.5783.  Google Scholar

[20]

J. X. Yin and C. H. Jin, Periodic solutions of the evolutionary p-Laplacian with nonlinear sources, J. Math. Anal. Appl., 368 (2010), 604-622. doi: 10.1016/j.jmaa.2010.03.006.  Google Scholar

show all references

References:
[1]

A. Beltramo and P. Hess, On the principal eigenvalue of a periodic-parabolic operator, Comm. Partial Differential Equations, 9 (1984), 919-941. doi: 10.1080/03605308408820351.  Google Scholar

[2]

H. Brezis and L. Veron, Removable singularities for some nonlinear elliptic equations, Arch. Ration. Mech. Anal., 75 (1980/1981), 1-6. doi: 10.1007/BF00284616.  Google Scholar

[3]

E. N. Dancer and P. Hess, On stable solutions of quasilinear periodic-parabolic problems, Ann. Scuola Norm. Sup. Pisa., 14 (1987), 123-141.  Google Scholar

[4]

M. J. Esteban, On periodic solutions of superlinear parabolic problems, Trans. Amer. Math. Soc., 293 (1986), 171-189. doi: 10.2307/2000278.  Google Scholar

[5]

M. J. Esteban, A remark on the existence of positive periodic solutions of superlinear parabolic problems, Proc. Amer. Math. Soc., 102 (1988), 131-136. doi: 10.2307/2046045.  Google Scholar

[6]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  Google Scholar

[7]

P. Lindqvist, On the equation div(|∇u|p-2u) + λ|u|p-2u = 0, Proc. Amer. Math. Soc., 109 (1990), 157-164. doi: 10.2307/2048375.  Google Scholar

[8]

V. Liskevich and I. I. Skrypnik, Isolated singularities of solutions to quasilinear elliptic equations, Potential Anal., 28 (2008), 1-16. doi: 10.1007/s11118-007-9063-3.  Google Scholar

[9]

V. Liskevich and I. I. Skrypnik, Isolated singularities of solutions to quasi-linear elliptic equations with absorption, J. Math. Anal. Appl., 338 (2008), 536-544. doi: 10.1016/j.jmaa.2007.05.018.  Google Scholar

[10]

N. Mizoguchi, Periodic solutions for degenerate diffusion equations, Indiana Univ. Math. J., 44 (1995), 413-432. doi: 10.1512/iumj.1995.44.1994.  Google Scholar

[11]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differencial Equations, Prentice Hall, Englewood Cliffs, 1967. (Chinese Trans. , Science Press, Beijing, 1985). Google Scholar

[12]

P. Quittner, Multiple equilibria, periodic solutions and a priori bounds for solutions in superlinear parabolic problems, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 237-258. doi: 10.1007/s00030-003-1056-3.  Google Scholar

[13]

T. I. Seidman, Periodic solutions of a non-linear parabolic equation, J. Differential Equations, 19 (1975), 242-257.  Google Scholar

[14]

J. Serrin, Local behaior of solutions of quasilinear equations, Acta Math., 111 (1964), 247-302.  Google Scholar

[15]

J. Serrin, Isolated singularities of solutions of quasilinear equations, Acta Math., 113 (1965), 219-240.  Google Scholar

[16]

J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math., 189 (2002), 79-142. doi: 10.1007/BF02392645.  Google Scholar

[17]

J. L. Vàzquez and L. Véron, Removable singularities of some strongly nonlinear elliptic equations, Manuscripta Math., 33 (1980), 129-144. doi: 10.1007/BF01316972.  Google Scholar

[18]

N. Mizoguchi, Periodic solutions for degenerate diffusion equations, Indiana Univ. Math. J., 44 (1995), 413-432. doi: 10.1512/iumj.1995.44.1994.  Google Scholar

[19]

Y. F. Wang, J. X. Yin and Z. Q. Wu, Periodic solutions of evolution p-Laplacian equations with nonlinear sources, J. Math. Anal. Appl., 219 (1998), 76-96. doi: 10.1006/jmaa.1997.5783.  Google Scholar

[20]

J. X. Yin and C. H. Jin, Periodic solutions of the evolutionary p-Laplacian with nonlinear sources, J. Math. Anal. Appl., 368 (2010), 604-622. doi: 10.1016/j.jmaa.2010.03.006.  Google Scholar

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