Positive radial solutions of a nonlinear boundary value problem

Patricio Cerda; Leonelo Iturriaga; Sebastián Lorca; Pedro Ubilla

doi:10.3934/cpaa.2018084

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September 2018, 17(5): 1765-1783. doi: 10.3934/cpaa.2018084

Positive radial solutions of a nonlinear boundary value problem

Patricio Cerda ^1, , Leonelo Iturriaga ^2, , Sebastián Lorca ^3, and Pedro Ubilla ^1,

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

Received May 2017 Revised January 2018 Published April 2018

In this work we study the following quasilinear elliptic equation:

$\left\{ {\begin{array}{*{20}{l}}{ - {\rm{div}}(\frac{{|x{|^\alpha }\nabla u}}{{{{(a(|x|) + g(u))}^\gamma }}}) = |x{|^\beta }{u^p}}&{{\rm{in}} \ \Omega }\\{u = 0}&{{\rm{on}}\;\;\;\;\partial \Omega }\end{array}} \right.$

where

$ a $

is a positive continuous function,

$ g $

is a nonnegative and nondecreasing continuous function,

$ Ω = B_R $

, is the ball of radius

$ R>0 $

centered at the origin in

$ \mathbb{R} ^N $

$N≥3 $

and, the constants

$ α,β∈\mathbb{R} $

$ γ∈(0,1) $

and

$ p>1 $

We derive a new Liouville type result for a kind of "broken equation". This result together with blow-up techniques, a priori estimates and a fixed-point result of Krasnosel'skii, allow us to ensure the existence of a positive radial solution. In this paper we also obtain a non-existence result, proven through a variation of the Pohozaev identity.

Keywords: Quasilinear elliptic equations, A priori estimates, Liouville theorems, fixed point theorems.

Mathematics Subject Classification: 35B09, 35B45, 35B53, 35J62, 58J20.

Citation: Patricio Cerda, Leonelo Iturriaga, Sebastián Lorca, Pedro Ubilla. Positive radial solutions of a nonlinear boundary value problem. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1765-1783. doi: 10.3934/cpaa.2018084

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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