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November  2019, 18(6): 3201-3216. doi: 10.3934/cpaa.2019144

## Existence theorem for a class of semilinear totally characteristic elliptic equations involving supercritical cone sobolev exponents

 School of Mathematics and Statistics, Wuhan University, Wuhan 330022, Hubei, China

*Corresponding author

Received  December 2018 Revised  December 2018 Published  May 2019

Fund Project: The authors are supported by NSFC grants 11771342 and 11571259

In this paper, we prove the existence of bounded positive solutions for a class of semilinear degenerate elliptic equations involving supercritical cone Sobolev exponents. We also obtain the existence of multiple solutions by the Ljusternik-Schnirelman theory.

Citation: Zhihua Huang, Xiaochun Liu. Existence theorem for a class of semilinear totally characteristic elliptic equations involving supercritical cone sobolev exponents. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3201-3216. doi: 10.3934/cpaa.2019144
##### References:
 [1] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Analysis, 14 (1976), 349-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar [2] Daomin Cao, Multiple solutions of a semilinear elliptic equation in $\mathbb{R}^N$, Ann. Inst. H. Poincaré, Analyse Nonlinéaire, 10 (1993), 593-604.  doi: 10.1016/S0294-1449(16)30198-6.  Google Scholar [3] J. Chabrowski, J. Yang, Existence theorems for elliptic equations involving supercritical Sobolev exponent, Advances in Differential Equations, 2 (1997), 231–256.  Google Scholar [4] H. Chen, X. Liu and Y. Wei, Existence theorem for a class of semilinear 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 [5] H. Chen, X. Liu and Y. Wei, Cone Sobolev inequality and Dirichlet problem for nonlinear elliptic equations on a manifold with conical singularities, Calc. Var. Partial Differ. Equ., 43 (2012), 463-484.  doi: 10.1007/s00526-011-0418-7.  Google Scholar [6] 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 [7] H. Chen, X. Liu and Y. Wei, Dirichlet problem for semilinear edge-degenerate elliptic equations with singular potential term, J. Differ. Equ., 252 (2012), 4289-4314.  doi: 10.1016/j.jde.2012.01.011.  Google Scholar [8] H. Chen, X. Liu and Y. Wei, Multiple solutions for semi-linear corner degenerate elliptic equations, J. Funct. Analysis, 266 (2014), 3815-3839.  doi: 10.1016/j.jfa.2013.12.012.  Google Scholar [9] H. Chen, R. Qiao, P. Luo and et al., Lower and upper bounds of Dirichlet eigenvalues for totally characteristic degenerate elliptic operators, Sci. China Math., 57 (2014), 2235-2246.  doi: 10.1007/s11425-014-4895-y.  Google Scholar [10] H. Chen, Y. Wei and B. Zhou, Existence of solutions for degenerate elliptic equations with singular potential on singular manifolds, Mathematische Nachrichten, 285 (2012), 1370–1384.  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] J. V. Egorov and B. W. Schulze, Pseudo-Differential Operators, Singularities, Applications, Operator Theory, Advances and Applications 93, Birkh$\ddot{\text{o}}$user, Basel, 1997. doi: 10.1007/978-3-0348-8900-1.  Google Scholar [13] H. Fan, Existence theorems for a class of edge-degenerate elliptic wquations on singular manifolds, Proc. Edinb. Math. Soc., 2015 (2015), 1-23.  doi: 10.1017/S0013091514000145.  Google Scholar [14] H. Fan, Multiple positive solutions for degenerate elliptic equations with singularity and critical cone Sobolev exponents, J. Pseudo-Differ. Oper. Appl., (2018). Google Scholar [15] H. Fan and X. Liu, Multiple positive solutions for degenerate elliptic equations with critical cone Sobolev exponents on singular manifolds, Electronic Journal of Differential Equations, 181 (2013), 1–22.  Google Scholar [16] X. Liu and S. Zhang, Multiple positive solutions for semi-linear elliptic systems involving sign-changing weight on manifolds with conical singularities, J. Pseudo-Differ. Oper. Appl., 7 (2016), 451-471.  doi: 10.1007/s11868-016-0147-y.  Google Scholar [17] V. A. Kondratev, Boundary value problems for elliptic equations in domains with conical points, Tr. Most. Mat. Obs., 16 (1967), 209–292.  Google Scholar [18] M. Lesch, Differential operators of Fuchs Type, Conical Singularities, and Asymptotic Methods, Teubner-Texte für Mathematik 136, Teubner-Verlag, Leipig, 1997.  Google Scholar [19] R. B. Melrose and G. A. Mendoza, Elliptic Operators of Totally Characteristic Type, Mathematical Science Research Institute, MSRI 047–83, 1983. Google Scholar [20] J. Moser, A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations, Communications on Pure and Applied Mathematics, 13 (1960), 457-468.  doi: 10.1002/cpa.3160130308.  Google Scholar [21] P. H. Rabinowitz, Variational methods for nonlinear elliptic eigenvalue problems, Indiana Univ. J. Maths, 23 (1974), 729-754.  doi: 10.1512/iumj.1974.23.23061.  Google Scholar [22] P. H. Rabinowitz, Minimax methods in critical points theory with applications to differential equation, CBMS Reg. Conf. Ser. Math, vol. 65 (Amer. Math. Soc), Providence, RI, 1986. doi: 10.1090/cbms/065.  Google Scholar [23] M. Willem, Minimax Theorem, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

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##### References:
 [1] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Analysis, 14 (1976), 349-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar [2] Daomin Cao, Multiple solutions of a semilinear elliptic equation in $\mathbb{R}^N$, Ann. Inst. H. Poincaré, Analyse Nonlinéaire, 10 (1993), 593-604.  doi: 10.1016/S0294-1449(16)30198-6.  Google Scholar [3] J. Chabrowski, J. Yang, Existence theorems for elliptic equations involving supercritical Sobolev exponent, Advances in Differential Equations, 2 (1997), 231–256.  Google Scholar [4] H. Chen, X. Liu and Y. Wei, Existence theorem for a class of semilinear 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 [5] H. Chen, X. Liu and Y. Wei, Cone Sobolev inequality and Dirichlet problem for nonlinear elliptic equations on a manifold with conical singularities, Calc. Var. Partial Differ. Equ., 43 (2012), 463-484.  doi: 10.1007/s00526-011-0418-7.  Google Scholar [6] 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 [7] H. Chen, X. Liu and Y. Wei, Dirichlet problem for semilinear edge-degenerate elliptic equations with singular potential term, J. Differ. Equ., 252 (2012), 4289-4314.  doi: 10.1016/j.jde.2012.01.011.  Google Scholar [8] H. Chen, X. Liu and Y. Wei, Multiple solutions for semi-linear corner degenerate elliptic equations, J. Funct. Analysis, 266 (2014), 3815-3839.  doi: 10.1016/j.jfa.2013.12.012.  Google Scholar [9] H. Chen, R. Qiao, P. Luo and et al., Lower and upper bounds of Dirichlet eigenvalues for totally characteristic degenerate elliptic operators, Sci. China Math., 57 (2014), 2235-2246.  doi: 10.1007/s11425-014-4895-y.  Google Scholar [10] H. Chen, Y. Wei and B. Zhou, Existence of solutions for degenerate elliptic equations with singular potential on singular manifolds, Mathematische Nachrichten, 285 (2012), 1370–1384.  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] J. V. Egorov and B. W. Schulze, Pseudo-Differential Operators, Singularities, Applications, Operator Theory, Advances and Applications 93, Birkh$\ddot{\text{o}}$user, Basel, 1997. doi: 10.1007/978-3-0348-8900-1.  Google Scholar [13] H. Fan, Existence theorems for a class of edge-degenerate elliptic wquations on singular manifolds, Proc. Edinb. Math. Soc., 2015 (2015), 1-23.  doi: 10.1017/S0013091514000145.  Google Scholar [14] H. Fan, Multiple positive solutions for degenerate elliptic equations with singularity and critical cone Sobolev exponents, J. Pseudo-Differ. Oper. Appl., (2018). Google Scholar [15] H. Fan and X. Liu, Multiple positive solutions for degenerate elliptic equations with critical cone Sobolev exponents on singular manifolds, Electronic Journal of Differential Equations, 181 (2013), 1–22.  Google Scholar [16] X. Liu and S. Zhang, Multiple positive solutions for semi-linear elliptic systems involving sign-changing weight on manifolds with conical singularities, J. Pseudo-Differ. Oper. Appl., 7 (2016), 451-471.  doi: 10.1007/s11868-016-0147-y.  Google Scholar [17] V. A. Kondratev, Boundary value problems for elliptic equations in domains with conical points, Tr. Most. Mat. Obs., 16 (1967), 209–292.  Google Scholar [18] M. Lesch, Differential operators of Fuchs Type, Conical Singularities, and Asymptotic Methods, Teubner-Texte für Mathematik 136, Teubner-Verlag, Leipig, 1997.  Google Scholar [19] R. B. Melrose and G. A. Mendoza, Elliptic Operators of Totally Characteristic Type, Mathematical Science Research Institute, MSRI 047–83, 1983. Google Scholar [20] J. Moser, A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations, Communications on Pure and Applied Mathematics, 13 (1960), 457-468.  doi: 10.1002/cpa.3160130308.  Google Scholar [21] P. H. Rabinowitz, Variational methods for nonlinear elliptic eigenvalue problems, Indiana Univ. J. Maths, 23 (1974), 729-754.  doi: 10.1512/iumj.1974.23.23061.  Google Scholar [22] P. H. Rabinowitz, Minimax methods in critical points theory with applications to differential equation, CBMS Reg. Conf. Ser. Math, vol. 65 (Amer. Math. Soc), Providence, RI, 1986. doi: 10.1090/cbms/065.  Google Scholar [23] M. Willem, Minimax Theorem, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar
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