Multiple solutions for nonlinear cone degenerate elliptic equations

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July & August 2021, 20(7&8): 2505-2518. 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 80^th birthday of Professor Shuxing Chen

Received May 2020 Revised September 2020 Published July & August 2021 Early access 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 and Applied Analysis, 2021, 20 (7&8) : 2505-2518. 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[12]	P. Drabek, Resonance Problems for the p -Laplacian, J Funct. Anal., 169 (1999), 189-200. doi: 10.1006/jfan.1999.3501.
[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.
[14]	D. A. Frank-Kamenetskii, Diffusion and Heat Transfer in Chemical Kinetics, Plenum Press, 1969.
[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.
[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.
[17]	R. B. Melrose and G. A. Mendoza, Elliptic operators of totally characteristic type, Math. Sci. Res., (1983), 29 pp.
[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.
[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.
[20]	B. W. Schulze, Boundary value problems and singular pseudo-differential operators, Pure Appl. Math., (1999).
[21]	H. Yamabe, On the deformations of Riemannian structures on compact manifolds, Osaka Math. J., (1960), 21–37.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[12]	P. Drabek, Resonance Problems for the p -Laplacian, J Funct. Anal., 169 (1999), 189-200. doi: 10.1006/jfan.1999.3501.
[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.
[14]	D. A. Frank-Kamenetskii, Diffusion and Heat Transfer in Chemical Kinetics, Plenum Press, 1969.
[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.
[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.
[17]	R. B. Melrose and G. A. Mendoza, Elliptic operators of totally characteristic type, Math. Sci. Res., (1983), 29 pp.
[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.
[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.
[20]	B. W. Schulze, Boundary value problems and singular pseudo-differential operators, Pure Appl. Math., (1999).
[21]	H. Yamabe, On the deformations of Riemannian structures on compact manifolds, Osaka Math. J., (1960), 21–37.

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