American Institute of Mathematical Sciences

Advanced
October  2014, 19(8): 2631-2639. doi: 10.3934/dcdsb.2014.19.2631

A note on the existence and properties of evanescent solutions for nonlinear elliptic problems

Aleksandra Orpel 1,

1. 

Faculty of Mathematics and Computer Science, University of Lodz, S. Banacha 22, 90-238 Lodz, Poland

Received  October 2013 Revised  April 2014 Published  August 2014

Basing ourselves on the subsolution and supersolution method we investigate the existence and properties of solutions of the following class of elliptic differential equations $div(a(||x||)\nabla u(x)) + f(x,u(x)) + g(||x||)k(x\cdot\nabla u(x)) = 0,$ $x\in\mathbb{R}^{n},||x||>R.$ Our main result concernes the behavior of solution at infinity.
Keywords: Positive evanescent solutions, nonlinear elliptic equation, asymptotic behavior., subsolution and supersolution method.
Mathematics Subject Classification: Primary: 35B40, 35J15, 35J60; Secondary: 34B15, 47H1.
Citation: Aleksandra Orpel. A note on the existence and properties of evanescent solutions for nonlinear elliptic problems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2631-2639. doi: 10.3934/dcdsb.2014.19.2631
References:
[1]

A. Constantin, Existence of positive solutions of quasilinear elliptic equations,, Bull. Austral. Math. Soc., 54 (1996), 147.  doi: 10.1017/S0004972700015148.  Google Scholar

[2]

A. Constantin, Positive solutions of quasilinear elliptic equations,, J. Math. Anal. Appl., 213 (1997), 334.  doi: 10.1006/jmaa.1997.5541.  Google Scholar

[3]

A. Constantin, On the existence of positive solutions of second order differential equations,, Ann. Mat. Pura Appl., 184 (2005), 131.  doi: 10.1007/s10231-004-0100-1.  Google Scholar

[4]

J. Deng, Bounded positive solutions of semilinear elliptic equations,, J. Math. Anal. Appl., 336 (2007), 1395.  doi: 10.1016/j.jmaa.2007.03.071.  Google Scholar

[5]

J. Deng, Existence of bounded positive solutions of semilinear elliptic equations,, Nonlin. Anal., 68 (2008), 3697.  doi: 10.1016/j.na.2007.04.012.  Google Scholar

[6]

S. Djebali, T. Moussaoui and O. G. Mustafa, Positive evanescent solutions of nonlinear elliptic equations,, J. Math. Anal. Appl., 333 (2007), 863.  doi: 10.1016/j.jmaa.2006.12.004.  Google Scholar

[7]

S. Djebali and A. Orpel, A note on positive evanescent solutions for a certain class of elliptic problems,, J Math. Anal. Appl., 353 (2009), 215.  doi: 10.1016/j.jmaa.2008.12.003.  Google Scholar

[8]

S. Djebali and A. Orpel, The continuous dependence on parameters of solutions for a class of elliptic problems on exterior domains,, Nonlinear Analysis, 73 (2010), 660.  doi: 10.1016/j.na.2010.03.054.  Google Scholar

[9]

M. Ehrnström, Positive solutions for second-order nonlinear differential equation,, Nonlinear Analysis, 64 (2006), 1608.  doi: 10.1016/j.na.2005.07.010.  Google Scholar

[10]

M. Ehrnström, On radial solutions of certain semi-linear elliptic equations,, Nonlinear Analysis, 64 (2006), 1578.  doi: 10.1016/j.na.2005.07.008.  Google Scholar

[11]

M. Ehrnström and O. G. Mustafa, On positive solutions of a class of nonlinear elliptic equations,, Nonlinear Analysis, 67 (2007), 1147.  doi: 10.1016/j.na.2006.07.002.  Google Scholar

[12]

E. S. Noussair and C. A. Swanson, Positive solutions of quasilinear elliptic equations in exterior domains,, J. Math. Anal. Appl., 75 (1980), 121.  doi: 10.1016/0022-247X(80)90310-8.  Google Scholar

[13]

B. Przeradzki and R. Stańczy, Positive solutions for sublinear elliptic equations,, Colloq. Math., 92 (2002), 141.  doi: 10.4064/cm92-1-12.  Google Scholar

[14]

E. Wahlén, Positive solutions of second-order differential equations,, Nonlinear Anal., 58 (2004), 359.  doi: 10.1016/j.na.2004.05.008.  Google Scholar

[15]

Z. Yin, Monotone positive solutions of second-order nonlinear differential equations,, Nonlinear Anal., 54 (2003), 391.  doi: 10.1016/S0362-546X(03)00089-0.  Google Scholar

show all references

References:
[1]

A. Constantin, Existence of positive solutions of quasilinear elliptic equations,, Bull. Austral. Math. Soc., 54 (1996), 147.  doi: 10.1017/S0004972700015148.  Google Scholar

[2]

A. Constantin, Positive solutions of quasilinear elliptic equations,, J. Math. Anal. Appl., 213 (1997), 334.  doi: 10.1006/jmaa.1997.5541.  Google Scholar

[3]

A. Constantin, On the existence of positive solutions of second order differential equations,, Ann. Mat. Pura Appl., 184 (2005), 131.  doi: 10.1007/s10231-004-0100-1.  Google Scholar

[4]

J. Deng, Bounded positive solutions of semilinear elliptic equations,, J. Math. Anal. Appl., 336 (2007), 1395.  doi: 10.1016/j.jmaa.2007.03.071.  Google Scholar

[5]

J. Deng, Existence of bounded positive solutions of semilinear elliptic equations,, Nonlin. Anal., 68 (2008), 3697.  doi: 10.1016/j.na.2007.04.012.  Google Scholar

[6]

S. Djebali, T. Moussaoui and O. G. Mustafa, Positive evanescent solutions of nonlinear elliptic equations,, J. Math. Anal. Appl., 333 (2007), 863.  doi: 10.1016/j.jmaa.2006.12.004.  Google Scholar

[7]

S. Djebali and A. Orpel, A note on positive evanescent solutions for a certain class of elliptic problems,, J Math. Anal. Appl., 353 (2009), 215.  doi: 10.1016/j.jmaa.2008.12.003.  Google Scholar

[8]

S. Djebali and A. Orpel, The continuous dependence on parameters of solutions for a class of elliptic problems on exterior domains,, Nonlinear Analysis, 73 (2010), 660.  doi: 10.1016/j.na.2010.03.054.  Google Scholar

[9]

M. Ehrnström, Positive solutions for second-order nonlinear differential equation,, Nonlinear Analysis, 64 (2006), 1608.  doi: 10.1016/j.na.2005.07.010.  Google Scholar

[10]

M. Ehrnström, On radial solutions of certain semi-linear elliptic equations,, Nonlinear Analysis, 64 (2006), 1578.  doi: 10.1016/j.na.2005.07.008.  Google Scholar

[11]

M. Ehrnström and O. G. Mustafa, On positive solutions of a class of nonlinear elliptic equations,, Nonlinear Analysis, 67 (2007), 1147.  doi: 10.1016/j.na.2006.07.002.  Google Scholar

[12]

E. S. Noussair and C. A. Swanson, Positive solutions of quasilinear elliptic equations in exterior domains,, J. Math. Anal. Appl., 75 (1980), 121.  doi: 10.1016/0022-247X(80)90310-8.  Google Scholar

[13]

B. Przeradzki and R. Stańczy, Positive solutions for sublinear elliptic equations,, Colloq. Math., 92 (2002), 141.  doi: 10.4064/cm92-1-12.  Google Scholar

[14]

E. Wahlén, Positive solutions of second-order differential equations,, Nonlinear Anal., 58 (2004), 359.  doi: 10.1016/j.na.2004.05.008.  Google Scholar

[15]

Z. Yin, Monotone positive solutions of second-order nonlinear differential equations,, Nonlinear Anal., 54 (2003), 391.  doi: 10.1016/S0362-546X(03)00089-0.  Google Scholar

[1]

Dumitru Motreanu, Calogero Vetro, Francesca Vetro. Systems of quasilinear elliptic equations with dependence on the gradient via subsolution-supersolution method. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 309-321. doi: 10.3934/dcdss.2018017
[2]

Yinbin Deng, Qi Gao. Asymptotic behavior of the positive solutions for an elliptic equation with Hardy term. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 367-380. doi: 10.3934/dcds.2009.24.367
[3]

Shinji Adachi, Masataka Shibata, Tatsuya Watanabe. Asymptotic behavior of positive solutions for a class of quasilinear elliptic equations with general nonlinearities. Communications on Pure & Applied Analysis, 2014, 13 (1) : 97-118. doi: 10.3934/cpaa.2014.13.97
[4]

Yongqin Liu. Asymptotic behavior of solutions to a nonlinear plate equation with memory. Communications on Pure & Applied Analysis, 2017, 16 (2) : 533-556. doi: 10.3934/cpaa.2017027
[5]

Limei Dai. Entire solutions with asymptotic behavior of fully nonlinear uniformly elliptic equations. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1707-1714. doi: 10.3934/cpaa.2011.10.1707
[6]

Nakao Hayashi, Pavel I. Naumkin. Asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation revisited. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 383-400. doi: 10.3934/dcds.1997.3.383
[7]

Xudong Shang, Jihui Zhang. Multi-peak positive solutions for a fractional nonlinear elliptic equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3183-3201. doi: 10.3934/dcds.2015.35.3183
[8]

Chunqing Lu. Asymptotic solutions of a nonlinear equation. Conference Publications, 2003, 2003 (Special) : 590-595. doi: 10.3934/proc.2003.2003.590
[9]

Jingyu Li. Asymptotic behavior of solutions to elliptic equations in a coated body. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1251-1267. doi: 10.3934/cpaa.2009.8.1251
[10]

Ling Mi. Asymptotic behavior for the unique positive solution to a singular elliptic problem. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1053-1072. doi: 10.3934/cpaa.2015.14.1053
[11]

Lie Zheng. Asymptotic behavior of solutions to the nonlinear breakage equations. Communications on Pure & Applied Analysis, 2005, 4 (2) : 463-473. doi: 10.3934/cpaa.2005.4.463
[12]

Irena Lasiecka, W. Heyman. Asymptotic behavior of solutions in nonlinear dynamic elasticity. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 237-252. doi: 10.3934/dcds.1995.1.237
[13]

Minkyu Kwak, Kyong Yu. The asymptotic behavior of solutions of a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - A, 1996, 2 (4) : 483-496. doi: 10.3934/dcds.1996.2.483
[14]

Shota Sato, Eiji Yanagida. Asymptotic behavior of singular solutions for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 4027-4043. doi: 10.3934/dcds.2012.32.4027
[15]

Tsung-Fang Wu. Multiplicity of positive solutions for a semilinear elliptic equation in $R_+^N$ with nonlinear boundary condition. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1675-1696. doi: 10.3934/cpaa.2010.9.1675
[16]

Goong Chen, Zhonghai Ding, Shujie Li. On positive solutions of the elliptic sine-Gordon equation. Communications on Pure & Applied Analysis, 2005, 4 (2) : 283-294. doi: 10.3934/cpaa.2005.4.283
[17]

Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $ \Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109
[18]

Long Wei. Concentrating phenomena in some elliptic Neumann problem: Asymptotic behavior of solutions. Communications on Pure & Applied Analysis, 2008, 7 (4) : 925-946. doi: 10.3934/cpaa.2008.7.925
[19]

Haitao Yang. On the existence and asymptotic behavior of large solutions for a semilinear elliptic problem in $R^n$. Communications on Pure & Applied Analysis, 2005, 4 (1) : 187-198. doi: 10.3934/cpaa.2005.4.197
[20]

Hongwei Zhang, Qingying Hu. Asymptotic behavior and nonexistence of wave equation with nonlinear boundary condition. Communications on Pure & Applied Analysis, 2005, 4 (4) : 861-869. doi: 10.3934/cpaa.2005.4.861

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences