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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 and 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), 147-154. doi: 10.1017/S0004972700015148.

[2]

A. Constantin, Positive solutions of quasilinear elliptic equations, J. Math. Anal. Appl., 213 (1997), 334-339. doi: 10.1006/jmaa.1997.5541.

[3]

A. Constantin, On the existence of positive solutions of second order differential equations, Ann. Mat. Pura Appl., 184 (2005), 131-138. doi: 10.1007/s10231-004-0100-1.

[4]

J. Deng, Bounded positive solutions of semilinear elliptic equations, J. Math. Anal. Appl., 336 (2007), 1395-1405. doi: 10.1016/j.jmaa.2007.03.071.

[5]

J. Deng, Existence of bounded positive solutions of semilinear elliptic equations, Nonlin. Anal., T.M.A., 68 (2008), 3697-3706. doi: 10.1016/j.na.2007.04.012.

[6]

S. Djebali, T. Moussaoui and O. G. Mustafa, Positive evanescent solutions of nonlinear elliptic equations, J. Math. Anal. Appl., 333 (2007), 863-870. doi: 10.1016/j.jmaa.2006.12.004.

[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), 215-223. doi: 10.1016/j.jmaa.2008.12.003.

[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), 660-672. doi: 10.1016/j.na.2010.03.054.

[9]

M. Ehrnström, Positive solutions for second-order nonlinear differential equation, Nonlinear Analysis, 64 (2006), 1608-1620. doi: 10.1016/j.na.2005.07.010.

[10]

M. Ehrnström, On radial solutions of certain semi-linear elliptic equations, Nonlinear Analysis, 64 (2006), 1578-1586. doi: 10.1016/j.na.2005.07.008.

[11]

M. Ehrnström and O. G. Mustafa, On positive solutions of a class of nonlinear elliptic equations, Nonlinear Analysis, 67 (2007), 1147-1154. doi: 10.1016/j.na.2006.07.002.

[12]

E. S. Noussair and C. A. Swanson, Positive solutions of quasilinear elliptic equations in exterior domains, J. Math. Anal. Appl., 75 (1980), 121-133. doi: 10.1016/0022-247X(80)90310-8.

[13]

B. Przeradzki and R. Stańczy, Positive solutions for sublinear elliptic equations, Colloq. Math., 92 (2002), 141-151. doi: 10.4064/cm92-1-12.

[14]

E. Wahlén, Positive solutions of second-order differential equations, Nonlinear Anal., 58 (2004), 359-366. doi: 10.1016/j.na.2004.05.008.

[15]

Z. Yin, Monotone positive solutions of second-order nonlinear differential equations, Nonlinear Anal., 54 (2003), 391-403. doi: 10.1016/S0362-546X(03)00089-0.

show all references

References:
[1]

A. Constantin, Existence of positive solutions of quasilinear elliptic equations, Bull. Austral. Math. Soc., 54 (1996), 147-154. doi: 10.1017/S0004972700015148.

[2]

A. Constantin, Positive solutions of quasilinear elliptic equations, J. Math. Anal. Appl., 213 (1997), 334-339. doi: 10.1006/jmaa.1997.5541.

[3]

A. Constantin, On the existence of positive solutions of second order differential equations, Ann. Mat. Pura Appl., 184 (2005), 131-138. doi: 10.1007/s10231-004-0100-1.

[4]

J. Deng, Bounded positive solutions of semilinear elliptic equations, J. Math. Anal. Appl., 336 (2007), 1395-1405. doi: 10.1016/j.jmaa.2007.03.071.

[5]

J. Deng, Existence of bounded positive solutions of semilinear elliptic equations, Nonlin. Anal., T.M.A., 68 (2008), 3697-3706. doi: 10.1016/j.na.2007.04.012.

[6]

S. Djebali, T. Moussaoui and O. G. Mustafa, Positive evanescent solutions of nonlinear elliptic equations, J. Math. Anal. Appl., 333 (2007), 863-870. doi: 10.1016/j.jmaa.2006.12.004.

[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), 215-223. doi: 10.1016/j.jmaa.2008.12.003.

[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), 660-672. doi: 10.1016/j.na.2010.03.054.

[9]

M. Ehrnström, Positive solutions for second-order nonlinear differential equation, Nonlinear Analysis, 64 (2006), 1608-1620. doi: 10.1016/j.na.2005.07.010.

[10]

M. Ehrnström, On radial solutions of certain semi-linear elliptic equations, Nonlinear Analysis, 64 (2006), 1578-1586. doi: 10.1016/j.na.2005.07.008.

[11]

M. Ehrnström and O. G. Mustafa, On positive solutions of a class of nonlinear elliptic equations, Nonlinear Analysis, 67 (2007), 1147-1154. doi: 10.1016/j.na.2006.07.002.

[12]

E. S. Noussair and C. A. Swanson, Positive solutions of quasilinear elliptic equations in exterior domains, J. Math. Anal. Appl., 75 (1980), 121-133. doi: 10.1016/0022-247X(80)90310-8.

[13]

B. Przeradzki and R. Stańczy, Positive solutions for sublinear elliptic equations, Colloq. Math., 92 (2002), 141-151. doi: 10.4064/cm92-1-12.

[14]

E. Wahlén, Positive solutions of second-order differential equations, Nonlinear Anal., 58 (2004), 359-366. doi: 10.1016/j.na.2004.05.008.

[15]

Z. Yin, Monotone positive solutions of second-order nonlinear differential equations, Nonlinear Anal., 54 (2003), 391-403. doi: 10.1016/S0362-546X(03)00089-0.

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