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Uniqueness for elliptic problems with Hölder--type dependence on the solution
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Uniqueness and comparison theorems for solutions of doubly nonlinear parabolic equations with nonstandard growth conditions
Keller-Osserman estimates for some quasilinear elliptic systems
1. | Laboratoire de Mathématiques et Physique Théorique, CNRS UMR 6083, Faculté des Sciences, 37200 Tours |
2. | Departamento de Matemáticas, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago |
3. | Departamento de Matemática y C.C., Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago |
References:
[1] |
M-F. Bidaut-Véron, Local and global behaviour of solutions of quasilinear equations of Emden-Fowler type, Arch. Rat. Mech. Anal., 107 (1989), 293-324.
doi: 10.1007/BF00251552. |
[2] |
M-F. Bidaut-Véron, Singularities of solutions of a class of quasilinear equations in divergence form, Nonlinear diffusion equations and their equilibrium states, Birkauser, Boston, Basel, Berlin, (1992), 129-144. |
[3] |
M-F. Bidaut-Véron, Removable singularities and existence for a quasilinear equation, Adv. Nonlinear Studies, 3 (2003), 25-63. |
[4] |
M-F. Bidaut-Véron and H. Giacomini, A new dynamical approach of Emden-Fowler equations and systems, Adv. Diff. Eq., 15 (2010), 1033-1082. |
[5] |
M-F. Bidaut-Véron and P. Grillot, Singularities in elliptic systems with absorption terms, Ann. Scuola Norm. Sup. Pisa CL. Sci, 28 (1999), 229-271. |
[6] |
M-F. Bidaut-Véron and P. Grillot, Asymptotic behaviour of elliptic systems with mixed absorption and source terms, Asymtotic Anal., 19 (1999), 117-147. |
[7] |
M-F. Bidaut-Véron, M. Garcia-Huidobro and C. Yarur, Large solutions of elliptic systems of second order and applications to the biharmonic equation, Discrete and Continuous Dynamical Systems, 32 (2012), 411-432. |
[8] |
M-F. Bidaut-Véron and S. Pohozaev, Nonexistence results and estimates for some nonlinear elliptic problems, J. Anal. Mathématique, 84 (2001), 1-49.
doi: 10.1007/BF02788105. |
[9] |
M-F. Bidaut-Véron and L. Véron, Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math., 106 (1991), 489-539.
doi: 10.1007/BF01243922. |
[10] |
L. d'Ambrosio and E. Mitidieri, A priori estimates, positivity results, and nonexistence theorems for quasilinear degenerate elliptic equations, Advances in Math., 224 (2010), 967-1020.
doi: 10.1016/j.aim.2009.12.017. |
[11] |
J. Dávila, L. Dupaigne, O. Goubet and S. Martinez, Boundary blow-up solutions of cooperative systems, Ann. I. H. Poincaré-AN, 26 (2009), 1767-1791. |
[12] |
E. Di Benedetto, "Partial Differential Equations," Birkaüser, 1995. |
[13] |
G. Dal Maso, F. Murat, L. Orsina and A. Prignet, Renormalized solutions of elliptic equations with general measure data, Ann. Scuola Norm. Sup. Pisa, 28 (1999), 741-808. |
[14] |
A. Farina and J. Serrin, Entire solutions of completely coercive quasilinear elliptic equations, J. Diff. Equ., 250 (2011), 4367-4408 and 4408-4436. |
[15] |
J. García-Melián, R. Letelier-Albornoz and J. Sabina de Lis, The solvability of an elliptic system under a singular boundary condition, Proc. Roy. Soc. Edinburgh, 136 (2006), 509-546. |
[16] |
J. García-Melian and J. Rossi, Boundary blow-up solutions to elliptic system of competitive type, J. Diff. Equ., 206 (2004), 156-181.
doi: 10.1016/j.jde.2003.12.004. |
[17] |
J. García-Melián, Large solutions for an elliptic system of quasilinear equations, J. Diff. Equ., 245 (2008), 3735-3752.
doi: 10.1016/j.jde.2008.04.004. |
[18] |
B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure and Applied Math., 34 (1981), 525-598.
doi: 10.1002/cpa.3160340406. |
[19] |
J. B. Keller, On the solutions of $-\Delta u=f(u),$ Comm. Pure Applied Math., 10 (1957), 503-510.
doi: 10.1002/cpa.3160100402. |
[20] |
T. Kilpelainen and J. Maly, Degenerate elliptic equations with measure data and non linear potentials, Ann. Scuola Norm. Sup. Pisa, 19 (1992), 591-613. |
[21] |
T. Kilpelainen and J. Maly, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Matematica, 172 (1994), 137-161.
doi: 10.1007/BF02392793. |
[22] |
T. Kilpelainen and X. Zhong, Growth of entire $\mathcal{A}$-subharmonic functions, Ann. Acad. Sci. Fennic Math., 28 (2003), 181-192. |
[23] |
E. Mitidieri and S. Pohozaev, Absence of positive solutions for quasilinear elliptic problems on $\mathbb{R}^N2$, Proc. Steklov Institute of Math., 227 (1999), 186-216. |
[24] |
R. Osserman, On the inequality $-\Delta u\geq f(u)$, Pacific J. Math., 7 (1957), 1641-1647. |
[25] |
J. Serrin, Local behavior of solutions of quasilinear equations, Acta Mathematica, 111 (1964), 247-302.
doi: 10.1007/BF02391014. |
[26] |
J. Serrin, Isolated singularities of solutions of quasilinear equations, Acta Mathematica, 113 (1965), 219-240.
doi: 10.1007/BF02391778. |
[27] |
J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Mathematica, 189 (2002), 79-142.
doi: 10.1007/BF02392645. |
[28] |
N. Trudinger, On Harnack type inequalities and their application to quasilinear equations, Comm. Pure Applied Math., 20 (1967), 721-747.
doi: 10.1002/cpa.3160200406. |
[29] |
J. L. Vazquez, An a priori interior estimate for the solutions of a nonlinear problem representing weak diffusion, Nonlinear Anal., 5 (1981), 95-103.
doi: 10.1016/0362-546X(81)90074-2. |
[30] |
J. L. Vazquez and L. Véron, Removable singularities of some strongly nonlinear elliptic equations, Manuscripta Math., 33 (1980), 129-144.
doi: 10.1007/BF01316972. |
[31] |
L. Véron, Semilinear elliptic equations with uniform blow-up on the boundary, J. Anal. Math., 59 (1992), 231-250.
doi: 10.1007/BF02790229. |
[32] |
M. Wu and Z. Yang, Existence of boundary blow-up solutions for a class of quasilinear elliptic systems with critical case, Applied Math. Comput., 198 (2008), 574-581.
doi: 10.1016/j.amc.2007.08.074. |
show all references
References:
[1] |
M-F. Bidaut-Véron, Local and global behaviour of solutions of quasilinear equations of Emden-Fowler type, Arch. Rat. Mech. Anal., 107 (1989), 293-324.
doi: 10.1007/BF00251552. |
[2] |
M-F. Bidaut-Véron, Singularities of solutions of a class of quasilinear equations in divergence form, Nonlinear diffusion equations and their equilibrium states, Birkauser, Boston, Basel, Berlin, (1992), 129-144. |
[3] |
M-F. Bidaut-Véron, Removable singularities and existence for a quasilinear equation, Adv. Nonlinear Studies, 3 (2003), 25-63. |
[4] |
M-F. Bidaut-Véron and H. Giacomini, A new dynamical approach of Emden-Fowler equations and systems, Adv. Diff. Eq., 15 (2010), 1033-1082. |
[5] |
M-F. Bidaut-Véron and P. Grillot, Singularities in elliptic systems with absorption terms, Ann. Scuola Norm. Sup. Pisa CL. Sci, 28 (1999), 229-271. |
[6] |
M-F. Bidaut-Véron and P. Grillot, Asymptotic behaviour of elliptic systems with mixed absorption and source terms, Asymtotic Anal., 19 (1999), 117-147. |
[7] |
M-F. Bidaut-Véron, M. Garcia-Huidobro and C. Yarur, Large solutions of elliptic systems of second order and applications to the biharmonic equation, Discrete and Continuous Dynamical Systems, 32 (2012), 411-432. |
[8] |
M-F. Bidaut-Véron and S. Pohozaev, Nonexistence results and estimates for some nonlinear elliptic problems, J. Anal. Mathématique, 84 (2001), 1-49.
doi: 10.1007/BF02788105. |
[9] |
M-F. Bidaut-Véron and L. Véron, Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math., 106 (1991), 489-539.
doi: 10.1007/BF01243922. |
[10] |
L. d'Ambrosio and E. Mitidieri, A priori estimates, positivity results, and nonexistence theorems for quasilinear degenerate elliptic equations, Advances in Math., 224 (2010), 967-1020.
doi: 10.1016/j.aim.2009.12.017. |
[11] |
J. Dávila, L. Dupaigne, O. Goubet and S. Martinez, Boundary blow-up solutions of cooperative systems, Ann. I. H. Poincaré-AN, 26 (2009), 1767-1791. |
[12] |
E. Di Benedetto, "Partial Differential Equations," Birkaüser, 1995. |
[13] |
G. Dal Maso, F. Murat, L. Orsina and A. Prignet, Renormalized solutions of elliptic equations with general measure data, Ann. Scuola Norm. Sup. Pisa, 28 (1999), 741-808. |
[14] |
A. Farina and J. Serrin, Entire solutions of completely coercive quasilinear elliptic equations, J. Diff. Equ., 250 (2011), 4367-4408 and 4408-4436. |
[15] |
J. García-Melián, R. Letelier-Albornoz and J. Sabina de Lis, The solvability of an elliptic system under a singular boundary condition, Proc. Roy. Soc. Edinburgh, 136 (2006), 509-546. |
[16] |
J. García-Melian and J. Rossi, Boundary blow-up solutions to elliptic system of competitive type, J. Diff. Equ., 206 (2004), 156-181.
doi: 10.1016/j.jde.2003.12.004. |
[17] |
J. García-Melián, Large solutions for an elliptic system of quasilinear equations, J. Diff. Equ., 245 (2008), 3735-3752.
doi: 10.1016/j.jde.2008.04.004. |
[18] |
B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure and Applied Math., 34 (1981), 525-598.
doi: 10.1002/cpa.3160340406. |
[19] |
J. B. Keller, On the solutions of $-\Delta u=f(u),$ Comm. Pure Applied Math., 10 (1957), 503-510.
doi: 10.1002/cpa.3160100402. |
[20] |
T. Kilpelainen and J. Maly, Degenerate elliptic equations with measure data and non linear potentials, Ann. Scuola Norm. Sup. Pisa, 19 (1992), 591-613. |
[21] |
T. Kilpelainen and J. Maly, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Matematica, 172 (1994), 137-161.
doi: 10.1007/BF02392793. |
[22] |
T. Kilpelainen and X. Zhong, Growth of entire $\mathcal{A}$-subharmonic functions, Ann. Acad. Sci. Fennic Math., 28 (2003), 181-192. |
[23] |
E. Mitidieri and S. Pohozaev, Absence of positive solutions for quasilinear elliptic problems on $\mathbb{R}^N2$, Proc. Steklov Institute of Math., 227 (1999), 186-216. |
[24] |
R. Osserman, On the inequality $-\Delta u\geq f(u)$, Pacific J. Math., 7 (1957), 1641-1647. |
[25] |
J. Serrin, Local behavior of solutions of quasilinear equations, Acta Mathematica, 111 (1964), 247-302.
doi: 10.1007/BF02391014. |
[26] |
J. Serrin, Isolated singularities of solutions of quasilinear equations, Acta Mathematica, 113 (1965), 219-240.
doi: 10.1007/BF02391778. |
[27] |
J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Mathematica, 189 (2002), 79-142.
doi: 10.1007/BF02392645. |
[28] |
N. Trudinger, On Harnack type inequalities and their application to quasilinear equations, Comm. Pure Applied Math., 20 (1967), 721-747.
doi: 10.1002/cpa.3160200406. |
[29] |
J. L. Vazquez, An a priori interior estimate for the solutions of a nonlinear problem representing weak diffusion, Nonlinear Anal., 5 (1981), 95-103.
doi: 10.1016/0362-546X(81)90074-2. |
[30] |
J. L. Vazquez and L. Véron, Removable singularities of some strongly nonlinear elliptic equations, Manuscripta Math., 33 (1980), 129-144.
doi: 10.1007/BF01316972. |
[31] |
L. Véron, Semilinear elliptic equations with uniform blow-up on the boundary, J. Anal. Math., 59 (1992), 231-250.
doi: 10.1007/BF02790229. |
[32] |
M. Wu and Z. Yang, Existence of boundary blow-up solutions for a class of quasilinear elliptic systems with critical case, Applied Math. Comput., 198 (2008), 574-581.
doi: 10.1016/j.amc.2007.08.074. |
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