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On existence and nonexistence of positive solutions of an elliptic system with coupled terms
Positive radial solutions of a nonlinear boundary value problem
1. | Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile |
2. | Universidad Técnica Federico Santa María, Av. Espana 1680, Casilla 110-V, Valparaíso, Chile |
3. | Instituto de Alta Investigación, Universidad de Tarapacá Casilla 7-D, Arica, Chile |
$\left\{ {\begin{array}{*{20}{l}}{ - {\rm{div}}(\frac{{|x{|^\alpha }\nabla u}}{{{{(a(|x|) + g(u))}^\gamma }}}) = |x{|^\beta }{u^p}}&{{\rm{in}} \ \Omega }\\{u = 0}&{{\rm{on}}\;\;\;\;\partial \Omega }\end{array}} \right.$ |
$ a $ |
$ g $ |
$ Ω = B_R $ |
$ R>0 $ |
$ \mathbb{R} ^N $ |
$N≥3 $ |
$ α,β∈\mathbb{R} $ |
$ γ∈(0,1) $ |
$ p>1 $ |
References:
[1] |
A. Alvino, L. Boccardo, V. Ferone, L. Orsina and G. Trombetti,
Existence results for nonlinear elliptic equations with degenerate coercivity, Annali di Matematica., 182 (2000), 53-79.
|
[2] |
A. Benkirane, A. Youssfi and D. Meskine,
Bounded solutions for nonlinear elliptic equations with degenerate coercivity and data in an L log L, Bull. Belg Math. Soc. Simon Stevin, 15 (2008), 369-375.
|
[3] |
L. Boccardo,
Some elliptic problems whit degenerate coercivity, Avanced Nonlinear Studies,, 6 (2006), 1-12.
|
[4] |
L. Boccardo and H. Brezis,
Some Remarks on a class of elliptic equations with degenerate coercivity, Bollettino U. M. I., 8 (2003), 521-530.
|
[5] |
L. Boccardo, A. Dall'aglio and L. Orsina,
Existence and regularity results for some elliptic equations with degenerate coercivity, Atti Sem. Mat. Fis. Univ. Modena., 46 (1998), suppl., 51-81.
|
[6] |
L. Boccardo, S. Segura de León and C. Trombetti,
Bounded and unbounded solutions for a class of quasi-linear elliptic problems whit a quadratic gradient term, J. Math. Pures Appl., 9 (2001), 919-940.
|
[7] |
P. Clement, D. de Figueiredo and E. Mitidieri,
Quasilinear elliptic equation with critical exponents, Topol. Methods Nonlinear Anal., 7 (1996), 133-170.
|
[8] |
P. Clement, R. Manásevich and E. Mitidieri,
Positive solutions for a quasilinear system via blow up, Comm. in P.D.E., 18 (1993), 2071-2106.
|
[9] |
L. Evans,
Partial Differential Equations, American Mathematical Soc., 01 June 1998. |
[10] |
M. A. Krasnosel'skii,
Positive Solutions of Operators Equations, Noordhoff, Groningen, 1964. |
[11] |
S. N. Armstrong and B. Sirakov,
Nonexistence of positive supersolutions of elliptic equations via the maximum principle, Comm. Partial Differential Equations, 36 (2011), 2011-2047.
|
[12] |
M-F. Bidaut-Veron,
Local and global behavior of solutions of quasilinear equations of Emden-Fowler type, Arch. Rational Mech. Anal., 107 (1989), 293-324.
|
[13] |
M-F. Bidaut-Veron and S. Pohozaev,
Nonexistence results and estimates for some nonlinear elliptic problems, J. Anal. Math., 84 (2001), 1-49.
|
[14] |
Ph. Clement, D. G. de Figueiredo and E. Mitidieri,
Positive solutions of semilinear elliptic systems, Comm. Partial Differential Equations, 17 (1992), 923-940.
|
[15] |
L. Damascelli, A. Farina, B. Sciunzi and E. Valdinoci, Liouville results for m-Laplace equations of Lane-Emden-Fowler type,
Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1099-1119 |
[16] |
B. Gidas and J. Spruck,
A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901.
|
[17] |
B. Gidas and J. Spruck,
J. Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.
|
[18] |
N. Kawano, W. Ni and S. Yotsutani,
A generalized Pohozaev identity and its applications, J. Math. Soc. Jpn., 42 (1990), 541-564.
|
[19] |
M. A. Krasnoselskii,
Fixed point of cone-compressing or cone-extending operators Soviet, Math. Dokl., 1 (1960), 1285-1288.
|
[20] |
E. Mitidieri and S. I. Pokhozhaev, Absence of positive solutions for quasilinear elliptic problems in $ \mathbb{R} ^N $, Tr. Mat. Inst. Steklova, 227 (1999) 192-222 (Issled. po Teor. Differ. Funkts. Mnogikh Perem. i ee Prilozh. 18). |
[21] |
S. I. Pohožaev,
On the eigenfunctions of the equation $ Δ u+λ f(u)=0 $, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39.
|
[22] |
J. Serrin and H. Zou,
Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math., 189 (2002), 79-142.
|
show all references
References:
[1] |
A. Alvino, L. Boccardo, V. Ferone, L. Orsina and G. Trombetti,
Existence results for nonlinear elliptic equations with degenerate coercivity, Annali di Matematica., 182 (2000), 53-79.
|
[2] |
A. Benkirane, A. Youssfi and D. Meskine,
Bounded solutions for nonlinear elliptic equations with degenerate coercivity and data in an L log L, Bull. Belg Math. Soc. Simon Stevin, 15 (2008), 369-375.
|
[3] |
L. Boccardo,
Some elliptic problems whit degenerate coercivity, Avanced Nonlinear Studies,, 6 (2006), 1-12.
|
[4] |
L. Boccardo and H. Brezis,
Some Remarks on a class of elliptic equations with degenerate coercivity, Bollettino U. M. I., 8 (2003), 521-530.
|
[5] |
L. Boccardo, A. Dall'aglio and L. Orsina,
Existence and regularity results for some elliptic equations with degenerate coercivity, Atti Sem. Mat. Fis. Univ. Modena., 46 (1998), suppl., 51-81.
|
[6] |
L. Boccardo, S. Segura de León and C. Trombetti,
Bounded and unbounded solutions for a class of quasi-linear elliptic problems whit a quadratic gradient term, J. Math. Pures Appl., 9 (2001), 919-940.
|
[7] |
P. Clement, D. de Figueiredo and E. Mitidieri,
Quasilinear elliptic equation with critical exponents, Topol. Methods Nonlinear Anal., 7 (1996), 133-170.
|
[8] |
P. Clement, R. Manásevich and E. Mitidieri,
Positive solutions for a quasilinear system via blow up, Comm. in P.D.E., 18 (1993), 2071-2106.
|
[9] |
L. Evans,
Partial Differential Equations, American Mathematical Soc., 01 June 1998. |
[10] |
M. A. Krasnosel'skii,
Positive Solutions of Operators Equations, Noordhoff, Groningen, 1964. |
[11] |
S. N. Armstrong and B. Sirakov,
Nonexistence of positive supersolutions of elliptic equations via the maximum principle, Comm. Partial Differential Equations, 36 (2011), 2011-2047.
|
[12] |
M-F. Bidaut-Veron,
Local and global behavior of solutions of quasilinear equations of Emden-Fowler type, Arch. Rational Mech. Anal., 107 (1989), 293-324.
|
[13] |
M-F. Bidaut-Veron and S. Pohozaev,
Nonexistence results and estimates for some nonlinear elliptic problems, J. Anal. Math., 84 (2001), 1-49.
|
[14] |
Ph. Clement, D. G. de Figueiredo and E. Mitidieri,
Positive solutions of semilinear elliptic systems, Comm. Partial Differential Equations, 17 (1992), 923-940.
|
[15] |
L. Damascelli, A. Farina, B. Sciunzi and E. Valdinoci, Liouville results for m-Laplace equations of Lane-Emden-Fowler type,
Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1099-1119 |
[16] |
B. Gidas and J. Spruck,
A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901.
|
[17] |
B. Gidas and J. Spruck,
J. Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.
|
[18] |
N. Kawano, W. Ni and S. Yotsutani,
A generalized Pohozaev identity and its applications, J. Math. Soc. Jpn., 42 (1990), 541-564.
|
[19] |
M. A. Krasnoselskii,
Fixed point of cone-compressing or cone-extending operators Soviet, Math. Dokl., 1 (1960), 1285-1288.
|
[20] |
E. Mitidieri and S. I. Pokhozhaev, Absence of positive solutions for quasilinear elliptic problems in $ \mathbb{R} ^N $, Tr. Mat. Inst. Steklova, 227 (1999) 192-222 (Issled. po Teor. Differ. Funkts. Mnogikh Perem. i ee Prilozh. 18). |
[21] |
S. I. Pohožaev,
On the eigenfunctions of the equation $ Δ u+λ f(u)=0 $, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39.
|
[22] |
J. Serrin and H. Zou,
Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math., 189 (2002), 79-142.
|
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