# American Institute of Mathematical Sciences

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 and Applied Analysis, 2017, 16 (2) : 373-392. doi: 10.3934/cpaa.2017019
##### 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. [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. [3] E. N. Dancer and P. Hess, On stable solutions of quasilinear periodic-parabolic problems, Ann. Scuola Norm. Sup. Pisa., 14 (1987), 123-141. [4] M. J. Esteban, On periodic solutions of superlinear parabolic problems, Trans. Amer. Math. Soc., 293 (1986), 171-189. doi: 10.2307/2000278. [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. [6] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. [7] P. Lindqvist, On the equation div(|∇u|p-2∇u) + λ|u|p-2u = 0, Proc. Amer. Math. Soc., 109 (1990), 157-164. doi: 10.2307/2048375. [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. [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. [10] N. Mizoguchi, Periodic solutions for degenerate diffusion equations, Indiana Univ. Math. J., 44 (1995), 413-432. doi: 10.1512/iumj.1995.44.1994. [11] M. H. Protter and H. F. Weinberger, Maximum Principles in Differencial Equations, Prentice Hall, Englewood Cliffs, 1967. (Chinese Trans. , Science Press, Beijing, 1985). [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. [13] T. I. Seidman, Periodic solutions of a non-linear parabolic equation, J. Differential Equations, 19 (1975), 242-257. [14] J. Serrin, Local behaior of solutions of quasilinear equations, Acta Math., 111 (1964), 247-302. [15] J. Serrin, Isolated singularities of solutions of quasilinear equations, Acta Math., 113 (1965), 219-240. [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. [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. [18] N. Mizoguchi, Periodic solutions for degenerate diffusion equations, Indiana Univ. Math. J., 44 (1995), 413-432. doi: 10.1512/iumj.1995.44.1994. [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. [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.

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##### 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. [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. [3] E. N. Dancer and P. Hess, On stable solutions of quasilinear periodic-parabolic problems, Ann. Scuola Norm. Sup. Pisa., 14 (1987), 123-141. [4] M. J. Esteban, On periodic solutions of superlinear parabolic problems, Trans. Amer. Math. Soc., 293 (1986), 171-189. doi: 10.2307/2000278. [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. [6] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. [7] P. Lindqvist, On the equation div(|∇u|p-2∇u) + λ|u|p-2u = 0, Proc. Amer. Math. Soc., 109 (1990), 157-164. doi: 10.2307/2048375. [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. [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. [10] N. Mizoguchi, Periodic solutions for degenerate diffusion equations, Indiana Univ. Math. J., 44 (1995), 413-432. doi: 10.1512/iumj.1995.44.1994. [11] M. H. Protter and H. F. Weinberger, Maximum Principles in Differencial Equations, Prentice Hall, Englewood Cliffs, 1967. (Chinese Trans. , Science Press, Beijing, 1985). [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. [13] T. I. Seidman, Periodic solutions of a non-linear parabolic equation, J. Differential Equations, 19 (1975), 242-257. [14] J. Serrin, Local behaior of solutions of quasilinear equations, Acta Math., 111 (1964), 247-302. [15] J. Serrin, Isolated singularities of solutions of quasilinear equations, Acta Math., 113 (1965), 219-240. [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. [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. [18] N. Mizoguchi, Periodic solutions for degenerate diffusion equations, Indiana Univ. Math. J., 44 (1995), 413-432. doi: 10.1512/iumj.1995.44.1994. [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. [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.
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