American Institute of Mathematical Sciences

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doi: 10.3934/cpaa.2020272

Multiple solutions for nonlinear cone degenerate elliptic equations

Hua Chen 1,, and  Yawei Wei 2,

1. 

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
2. 

School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, China

* Corresponding author

Dedicated to the 80th birthday of Professor Shuxing Chen

Received  May 2020 Revised  September 2020 Published  November 2020

Fund Project: This work is supported by the NSFC under the grands 11771218, 11371282, 11631011 and supported by the Fundamental Research Funds for the Central Universities

The present paper is concerned with the Dirichlet boundary value problem for nonlinear cone degenerate elliptic equations. First we introduce the weighted Sobolev spaces, inequalities and the property of compactness. After the appropriate energy functional established, we obtain the existence of infinitely many solutions in the weighted Sobolev spaces by applying the variational methods.

Keywords: Degenerate operator, conical singularity, variational methods, weighted Sobolev spaces.
Mathematics Subject Classification: Primary: 35J20; Secondary: 35J70.
Citation: Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2020272
References:
[1]

R. P. Agarwal, M. B. Ghaemi and S. Saiedinezhad, The Nehari manifold for the degenerate p-Laplacian quasilinear elliptic equations, Adv. Math. Sci. Appl., 20 (2010), 37-50.   Google Scholar

[2]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Func. Anal., 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[3]

D. Cao, S. Peng and S. Yan, Infinitely many solutions for p-Laplacian equation involving critical Sobolev growth, J. Func. Anal., 262 (2012), 2861-2902.  doi: 10.1016/j.jfa.2012.01.006.  Google Scholar

[4]

S. Carl and D. Motreanu, Multiple and sign-changing solutions for the multivalued p-Laplacian equation, Math. Nachr., 283 (2010), 965-981.  doi: 10.1002/mana.200710049.  Google Scholar

[5]

A. Cavalheiro, Existence results for Dirichlet problems with degenerated p-Laplacian and p-biharmonic operators, Opuscula Math., 33 (2013), 439-453. doi: 10.7494/OpMath.2013.33.3.439.  Google Scholar

[6]

A. Cavalheiro, Existence and Uniqueness of Solutions for Dirichlet Problems with Degenerate Nonlinear Elliptic Operators, Differ. Equ. Dyn. Syst., 24 (2016), 305-317. doi: 10.1007/s12591-014-0214-x.  Google Scholar

[7]

H. Chen, X. Liu and Y. Wei, Existence Theorem for a class of Semi-linear totally Characteristic Elliptic Equations with Critical Cone Sobolev Exponents, Ann. Glob. Anal. Geom., 39 (2011), 27-43.  doi: 10.1007/s10455-010-9226-0.  Google Scholar

[8]

H. Chen, X. Liu and Y. Wei, Cone Sobolev Inequality and Dirichlet problems for Nonlinear Elliptic Equations on Manifold with Conical Singularities, Calc. Var. PDEs, 43 (2012), 463-484.  doi: 10.1007/s00526-011-0418-7.  Google Scholar

[9]

H. Chen, X. Liu and Y. Wei, Multiple Solutions for Semilinear totally Characteristic Elliptic Equations with Subcritical or Critical Cone Sobolev Exponents, J. Differ. Equ., 252 (2012), 4200-4228.  doi: 10.1016/j.jde.2011.12.009.  Google Scholar

[10]

H. Chen, Y. Wei and B. Zhou, Existence of Solutions for Degenerate Elliptic Equations with Singular Potential on Conical Singular Manifolds, Math. Nachr., 285 (2012), 1370-1384.  doi: 10.1002/mana.201100088.  Google Scholar

[11]

S. Coriasco, E. Schrohe and J. Seiler, Realizations of differential operators on conic manifolds with boundary, Ann. Glob. Anal. Geom., 31 (2007), 223-285.  doi: 10.1007/s10455-006-9019-7.  Google Scholar

[12]

P. Drabek, Resonance Problems for the p -Laplacian, J Funct. Anal., 169 (1999), 189-200.  doi: 10.1006/jfan.1999.3501.  Google Scholar

[13]

Ju. V. Egorov and B. W. Schulze, Pseudo-differential operators, singularities, applications, Operator Theory, Advances and Applications 93, Birkhäuser Verlag, Basel, 1997. doi: 10.1007/978-3-0348-8900-1.  Google Scholar

[14] D. A. Frank-Kamenetskii, Diffusion and Heat Transfer in Chemical Kinetics, Plenum Press, 1969.   Google Scholar
[15]

J. Garcia Azorero and I. Peral Alonso, Existence and nonuniqueness for the $p$-Laplacian: Nonlinear Eigenvalues, Commun. in PDE, 12 (1987), 1389-1430.  doi: 10.1080/03605308708820534.  Google Scholar

[16]

Y. Jing and Z. Liu, Infinitely many solutions of p-sublinear p-Laplacian equations, J. Math. Anal. Appl., 429 (2015), 1240-1257.  doi: 10.1016/j.jmaa.2015.04.069.  Google Scholar

[17]

R. B. Melrose and G. A. Mendoza, Elliptic operators of totally characteristic type, Math. Sci. Res., (1983), 29 pp. Google Scholar

[18]

P. H. Rabinowitz, Some Aspects of Nonlinear Eigenvalue Problems, Rocky Mt. J. Math., 2 (1973), 161-192.  doi: 10.1216/RMJ-1973-3-2-161.  Google Scholar

[19]

E. Schrohe and J. Seiler, Ellipticity and invertibility in the cone algebra on $L_p$-Sobolev spaces, Integr. Equat. Oper. Th., (2001), 93–114. doi: 10.1007/BF01202533.  Google Scholar

[20]

B. W. Schulze, Boundary value problems and singular pseudo-differential operators, Pure Appl. Math., (1999).  Google Scholar

[21]

H. Yamabe, On the deformations of Riemannian structures on compact manifolds, Osaka Math. J., (1960), 21–37.  Google Scholar

show all references

References:
[1]

R. P. Agarwal, M. B. Ghaemi and S. Saiedinezhad, The Nehari manifold for the degenerate p-Laplacian quasilinear elliptic equations, Adv. Math. Sci. Appl., 20 (2010), 37-50.   Google Scholar

[2]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Func. Anal., 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[3]

D. Cao, S. Peng and S. Yan, Infinitely many solutions for p-Laplacian equation involving critical Sobolev growth, J. Func. Anal., 262 (2012), 2861-2902.  doi: 10.1016/j.jfa.2012.01.006.  Google Scholar

[4]

S. Carl and D. Motreanu, Multiple and sign-changing solutions for the multivalued p-Laplacian equation, Math. Nachr., 283 (2010), 965-981.  doi: 10.1002/mana.200710049.  Google Scholar

[5]

A. Cavalheiro, Existence results for Dirichlet problems with degenerated p-Laplacian and p-biharmonic operators, Opuscula Math., 33 (2013), 439-453. doi: 10.7494/OpMath.2013.33.3.439.  Google Scholar

[6]

A. Cavalheiro, Existence and Uniqueness of Solutions for Dirichlet Problems with Degenerate Nonlinear Elliptic Operators, Differ. Equ. Dyn. Syst., 24 (2016), 305-317. doi: 10.1007/s12591-014-0214-x.  Google Scholar

[7]

H. Chen, X. Liu and Y. Wei, Existence Theorem for a class of Semi-linear totally Characteristic Elliptic Equations with Critical Cone Sobolev Exponents, Ann. Glob. Anal. Geom., 39 (2011), 27-43.  doi: 10.1007/s10455-010-9226-0.  Google Scholar

[8]

H. Chen, X. Liu and Y. Wei, Cone Sobolev Inequality and Dirichlet problems for Nonlinear Elliptic Equations on Manifold with Conical Singularities, Calc. Var. PDEs, 43 (2012), 463-484.  doi: 10.1007/s00526-011-0418-7.  Google Scholar

[9]

H. Chen, X. Liu and Y. Wei, Multiple Solutions for Semilinear totally Characteristic Elliptic Equations with Subcritical or Critical Cone Sobolev Exponents, J. Differ. Equ., 252 (2012), 4200-4228.  doi: 10.1016/j.jde.2011.12.009.  Google Scholar

[10]

H. Chen, Y. Wei and B. Zhou, Existence of Solutions for Degenerate Elliptic Equations with Singular Potential on Conical Singular Manifolds, Math. Nachr., 285 (2012), 1370-1384.  doi: 10.1002/mana.201100088.  Google Scholar

[11]

S. Coriasco, E. Schrohe and J. Seiler, Realizations of differential operators on conic manifolds with boundary, Ann. Glob. Anal. Geom., 31 (2007), 223-285.  doi: 10.1007/s10455-006-9019-7.  Google Scholar

[12]

P. Drabek, Resonance Problems for the p -Laplacian, J Funct. Anal., 169 (1999), 189-200.  doi: 10.1006/jfan.1999.3501.  Google Scholar

[13]

Ju. V. Egorov and B. W. Schulze, Pseudo-differential operators, singularities, applications, Operator Theory, Advances and Applications 93, Birkhäuser Verlag, Basel, 1997. doi: 10.1007/978-3-0348-8900-1.  Google Scholar

[14] D. A. Frank-Kamenetskii, Diffusion and Heat Transfer in Chemical Kinetics, Plenum Press, 1969.   Google Scholar
[15]

J. Garcia Azorero and I. Peral Alonso, Existence and nonuniqueness for the $p$-Laplacian: Nonlinear Eigenvalues, Commun. in PDE, 12 (1987), 1389-1430.  doi: 10.1080/03605308708820534.  Google Scholar

[16]

Y. Jing and Z. Liu, Infinitely many solutions of p-sublinear p-Laplacian equations, J. Math. Anal. Appl., 429 (2015), 1240-1257.  doi: 10.1016/j.jmaa.2015.04.069.  Google Scholar

[17]

R. B. Melrose and G. A. Mendoza, Elliptic operators of totally characteristic type, Math. Sci. Res., (1983), 29 pp. Google Scholar

[18]

P. H. Rabinowitz, Some Aspects of Nonlinear Eigenvalue Problems, Rocky Mt. J. Math., 2 (1973), 161-192.  doi: 10.1216/RMJ-1973-3-2-161.  Google Scholar

[19]

E. Schrohe and J. Seiler, Ellipticity and invertibility in the cone algebra on $L_p$-Sobolev spaces, Integr. Equat. Oper. Th., (2001), 93–114. doi: 10.1007/BF01202533.  Google Scholar

[20]

B. W. Schulze, Boundary value problems and singular pseudo-differential operators, Pure Appl. Math., (1999).  Google Scholar

[21]

H. Yamabe, On the deformations of Riemannian structures on compact manifolds, Osaka Math. J., (1960), 21–37.  Google Scholar

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