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May  2018, 38(5): 2411-2439. doi: 10.3934/dcds.2018100

Positive periodic solutions of the weighted $p$-Laplacian with nonlinear sources

1. 

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

2. 

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

3. 

School of Science and Engineering, Chinese University of Hong Kong, Shenzhen, Guangdong 518172, China

* Corresponding author: J. Yin

Received  August 2017 Revised  January 2018 Published  March 2018

Fund Project: The first author is supported by the Fundamental Research Funds for the Central Universities (No. 2017BQ109), China Postdoctoral Science Foundation (No. 2017M610517) and NSFC Grant No. 11701184. The second author is supported by NSFC Grant No. 11771156.

In this paper we study the existence of time periodic solutions for the evolutionary weighted $p$-Laplacian with a nonlinear periodic source in a bounded domain containing the origin. We show that there is a critical exponent $q_c = q_c(α,β) = \frac{(N+β)p}{N+α-p}-1$ and a singular exponent $q_s = p-1$: there exists a positive periodic solution when $0<q<q_c$ and $q\ne q_s$; while there is no positive periodic solution when $q≥ q_c$. The case when $q = q_s$ is completely different from the remaining case $q\ne q_s$, the problem may or may not have solutions depending on the coefficients of the equation.

Citation: Shanming Ji, Jingxue Yin, Yutian Li. Positive periodic solutions of the weighted $p$-Laplacian with nonlinear sources. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2411-2439. doi: 10.3934/dcds.2018100
References:
[1]

C. Azizieh and P. Clément, A priori estimates and continuation methods for positive solutions of $p$-Laplace equations, J. Differential Equations, 179 (2002), 213-245.  doi: 10.1006/jdeq.2001.4029.  Google Scholar

[2]

A. Beltramo, Über den Haupteigenwert von periodisch-parabolischen differential operatoren, Ph. D. Thesis, University of Zürich, 1984. Google Scholar

[3]

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

[4]

M.-F. Bidaut-Véron and M. García-Huidobro, Regular and singular solutions of a quasilinear equation with weights, Asymptotic Analysis, 28 (2001), 115-150.   Google Scholar

[5]

D. Daners, Periodic-parabolic eigenvalue problems with indefinite weight functions, Arch. Math., 68 (1997), 388-397.  doi: 10.1007/s000130050071.  Google Scholar

[6]

M. J. Esteban, On periodic solutions of superlinear parabolic problems, Trans. Amer. Math. Soc., 293 (1986), 171-189.  doi: 10.1090/S0002-9947-1986-0814919-8.  Google Scholar

[7]

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: 10.1090/S0002-9939-1988-0915730-7.  Google Scholar

[8]

B. GidasW. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  doi: 10.1007/BF01221125.  Google Scholar

[9]

T. Godoy and U. Kaufmann, On principal eigenvalues for periodic parabolic problems with optimal condition on the weight function, J. Math. Anal. Appl., 262 (2001), 208-220.  doi: 10.1006/jmaa.2001.7559.  Google Scholar

[10]

T. GodoyE. Lami Dozo and S. Paczka, The periodic parabolic eigenvalue problem with $L^∞$ weight, Math. Scand., 81 (1997), 20-34.  doi: 10.7146/math.scand.a-12864.  Google Scholar

[11]

S. M. JiY. T. LiR. Huang and J. X. Yin, Singular periodic solutions for the $p$-Laplacian in a punctured domain, Comm. Pure Appl. Anal., 16 (2017), 373-392.  doi: 10.3934/cpaa.2017019.  Google Scholar

[12]

S. M. JiJ. X. Yin and R. Huang, Oscillatory traveling waves of polytropic filtration equation with generalized Fisher-KPP sources, J. Math. Anal. Appl., 419 (2014), 68-78.  doi: 10.1016/j.jmaa.2014.04.030.  Google Scholar

[13]

U. Kaufmann, Some results on principal eigenvalues for periodic parabolic problems with weight, Bull. Austral. Math. Soc., 68 (2003), 177-184.  doi: 10.1017/S0004972700037564.  Google Scholar

[14]

Y. X. Li and C. H. Xie, Blow-up for $p$-Laplacian parabolic equations, Electron. J. Differential Equations, 20 (2003), 1-12.   Google Scholar

[15]

P. Lindqvist, On the equation $\mbox{div}(|\nabla u|^{p-2}\nabla u)+λ|u|^{p-2}u = 0$, Proc. Amer. Math. Soc., 109 (1990), 157-164.  doi: 10.1090/S0002-9939-1990-1007505-7.  Google Scholar

[16]

E. Mitidieri and S. I. Pohozaev, Nonexistence of positive solutions for quasilinear elliptic problems on $\mathbb R^N$, Proc. Steklov Inst. Math., 227 (1999), 186-216.   Google Scholar

[17]

N. Mizoguchi, Periodic solutions for degenerate diffusion equations, Indiana Univ. Math. J., 44 (1995), 413-432.   Google Scholar

[18]

W. M. Ni and J. Serrin, Nonexistence theorems for quasilinear partial differential equations, Rend. Circ. Mat. Palermo (2) Suppl., 8 (1985), 171-185.   Google Scholar

[19]

M. Ôtani, Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations, J. Funct. Anal., 76 (1988), 140-159.  doi: 10.1016/0022-1236(88)90053-5.  Google Scholar

[20]

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.   Google Scholar

[21]

E. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, American Journal of Mathematics, 114 (1992), 813-874.  doi: 10.2307/2374799.  Google Scholar

[22]

T. I. Seidman, Periodic solutions of a non-linear parabolic equation, J. Differential Equations, 19 (1975), 242-257.  doi: 10.1016/0022-0396(75)90004-2.  Google Scholar

[23]

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

[24]

H. Song, J. Yin and Z. Wang, Isolated singularities of positive solutions to the weighted $p$-Laplacian, Calc. Var., 55 (2016), Art. 28, 16 pp.  Google Scholar

[25]

Y. F. WangJ. 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

[26]

Z. Q. Wu, J. X. Yin and C. P. Wang, Elliptic and Parabolic Equations, World Scientific, 2006. doi: 10.1142/6238.  Google Scholar

[27]

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]

C. Azizieh and P. Clément, A priori estimates and continuation methods for positive solutions of $p$-Laplace equations, J. Differential Equations, 179 (2002), 213-245.  doi: 10.1006/jdeq.2001.4029.  Google Scholar

[2]

A. Beltramo, Über den Haupteigenwert von periodisch-parabolischen differential operatoren, Ph. D. Thesis, University of Zürich, 1984. Google Scholar

[3]

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

[4]

M.-F. Bidaut-Véron and M. García-Huidobro, Regular and singular solutions of a quasilinear equation with weights, Asymptotic Analysis, 28 (2001), 115-150.   Google Scholar

[5]

D. Daners, Periodic-parabolic eigenvalue problems with indefinite weight functions, Arch. Math., 68 (1997), 388-397.  doi: 10.1007/s000130050071.  Google Scholar

[6]

M. J. Esteban, On periodic solutions of superlinear parabolic problems, Trans. Amer. Math. Soc., 293 (1986), 171-189.  doi: 10.1090/S0002-9947-1986-0814919-8.  Google Scholar

[7]

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: 10.1090/S0002-9939-1988-0915730-7.  Google Scholar

[8]

B. GidasW. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  doi: 10.1007/BF01221125.  Google Scholar

[9]

T. Godoy and U. Kaufmann, On principal eigenvalues for periodic parabolic problems with optimal condition on the weight function, J. Math. Anal. Appl., 262 (2001), 208-220.  doi: 10.1006/jmaa.2001.7559.  Google Scholar

[10]

T. GodoyE. Lami Dozo and S. Paczka, The periodic parabolic eigenvalue problem with $L^∞$ weight, Math. Scand., 81 (1997), 20-34.  doi: 10.7146/math.scand.a-12864.  Google Scholar

[11]

S. M. JiY. T. LiR. Huang and J. X. Yin, Singular periodic solutions for the $p$-Laplacian in a punctured domain, Comm. Pure Appl. Anal., 16 (2017), 373-392.  doi: 10.3934/cpaa.2017019.  Google Scholar

[12]

S. M. JiJ. X. Yin and R. Huang, Oscillatory traveling waves of polytropic filtration equation with generalized Fisher-KPP sources, J. Math. Anal. Appl., 419 (2014), 68-78.  doi: 10.1016/j.jmaa.2014.04.030.  Google Scholar

[13]

U. Kaufmann, Some results on principal eigenvalues for periodic parabolic problems with weight, Bull. Austral. Math. Soc., 68 (2003), 177-184.  doi: 10.1017/S0004972700037564.  Google Scholar

[14]

Y. X. Li and C. H. Xie, Blow-up for $p$-Laplacian parabolic equations, Electron. J. Differential Equations, 20 (2003), 1-12.   Google Scholar

[15]

P. Lindqvist, On the equation $\mbox{div}(|\nabla u|^{p-2}\nabla u)+λ|u|^{p-2}u = 0$, Proc. Amer. Math. Soc., 109 (1990), 157-164.  doi: 10.1090/S0002-9939-1990-1007505-7.  Google Scholar

[16]

E. Mitidieri and S. I. Pohozaev, Nonexistence of positive solutions for quasilinear elliptic problems on $\mathbb R^N$, Proc. Steklov Inst. Math., 227 (1999), 186-216.   Google Scholar

[17]

N. Mizoguchi, Periodic solutions for degenerate diffusion equations, Indiana Univ. Math. J., 44 (1995), 413-432.   Google Scholar

[18]

W. M. Ni and J. Serrin, Nonexistence theorems for quasilinear partial differential equations, Rend. Circ. Mat. Palermo (2) Suppl., 8 (1985), 171-185.   Google Scholar

[19]

M. Ôtani, Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations, J. Funct. Anal., 76 (1988), 140-159.  doi: 10.1016/0022-1236(88)90053-5.  Google Scholar

[20]

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.   Google Scholar

[21]

E. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, American Journal of Mathematics, 114 (1992), 813-874.  doi: 10.2307/2374799.  Google Scholar

[22]

T. I. Seidman, Periodic solutions of a non-linear parabolic equation, J. Differential Equations, 19 (1975), 242-257.  doi: 10.1016/0022-0396(75)90004-2.  Google Scholar

[23]

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

[24]

H. Song, J. Yin and Z. Wang, Isolated singularities of positive solutions to the weighted $p$-Laplacian, Calc. Var., 55 (2016), Art. 28, 16 pp.  Google Scholar

[25]

Y. F. WangJ. 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

[26]

Z. Q. Wu, J. X. Yin and C. P. Wang, Elliptic and Parabolic Equations, World Scientific, 2006. doi: 10.1142/6238.  Google Scholar

[27]

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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