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Analysis of positive solutions for a class of semipositone p-Laplacian problems with nonlinear boundary conditions
1. | Department of Mathematics Education, Pusan National University, Busan 46241, Korea |
2. | Department of Mathematics and Statistics, University of North Carolina at Greensboro, Greensboro, NC 27412, USA |
3. | Department of Mathematics, University of Ulsan, Ulsan 44610, Korea |
4. | Department of Mathematics, Wayne State University, Detroit, MI 48202, USA |
$\left\{ \begin{align} & -\left( {{\varphi }_{p}}(u') \right)'=\lambda h(t)\frac{f(u)}{{{u}^{\alpha }}},~\ \ t\in (0,1),~~ \\ & u'(1)+c(u(1))u(1)=0,~ \\ & u(0)=0, \\ \end{align} \right.$ |
$\varphi_p(u): = |u|^{p-2}u$ |
$p>1$ |
$p$ |
$u$ |
$λ>0$ |
$0≤α<1$ |
$c:[0,∞)\rightarrow (0,∞)$ |
$h:(0,1)\rightarrow (0,∞)$ |
$f∈ C[0,∞)$ |
$f(0)<0$ |
$\lim_{s\rightarrow ∞}f(s) = ∞$ |
$\frac{f(s)}{s^{α}}$ |
$p$ |
$\lim_{s \rightarrow ∞}\frac{f(s)}{s^{p-1+α}} = 0$ |
$λ≈ 0$ |
$λ \gg 1$ |
References:
[1] |
H. Amann,
Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709.
doi: 10.1137/1018114. |
[2] |
D. Butler, E. Ko, E. K. Lee and R. Shivaji,
Positive radial solutions for elliptic equations on exterior domains with nonlinear boundary conditions, Commun. Pure Appl. Anal., 13 (2014), 2713-2731.
doi: 10.3934/cpaa.2014.13.2713. |
[3] |
R. S. Cantrell and C. Cosner,
Density dependent behavior at habitat boundaries and the allee effect, Bull. Math. Biol., 69 (2007), 2339-2360.
doi: 10.1007/s11538-007-9222-0. |
[4] |
R. S. Cantrell and C. Cosner, Spatial ecology via reaction-diffusion equations, John Wiley &
Sons, Chichester, 2004.
doi: 10.1002/0470871296. |
[5] |
D. Daners,
Robin boundary value problems on arbitrary domains, Trans. Amer. Math. Soc., 352 (2000), 4207-4236.
doi: 10.1090/S0002-9947-00-02444-2. |
[6] |
D. A. Frank-Kamenetskii, Diffusion and Heat Transfer in Chemical Kinetics, New York, Plenum Press, 1969.
![]() |
[7] |
J. Goddard II, E. K. Lee and R. Shivaji,
Population models with diffusion, strong allee effect, and nonlinear boundary conditions, Nonlinear Anal., 74 (2011), 6202-6208.
doi: 10.1016/j.na.2011.06.001. |
[8] |
D. D. Hai,
Uniqueness of positive solutions for a class of quasilinear problems, Nonlinear Anal., 69 (2008), 2720-2732.
doi: 10.1016/j.na.2007.08.046. |
[9] |
E. Ko, M. Ramaswamy and R. Shivaji,
Uniqueness of positive radial solutions for a class of semipositone problems on the exterior of a ball, J. Math. Anal. Appl., 423 (2015), 399-409.
doi: 10.1016/j.jmaa.2014.09.058. |
[10] |
E. K. Lee, R. Shivaji and B. Son,
Positive radial solutions to classes of singular problems on the exterior domain of a ball, J. Math. Anal. Appl., 434 (2016), 1597-1611.
doi: 10.1016/j.jmaa.2015.09.072. |
[11] |
P. Drábek, Topological and Variational Methods for Nonlinear Boundary Value Problems,
1st edition, Addison Wesley Longman Limited, Harlow, 1997 |
[12] |
M. D. Pino, M. Elgueta and R. Manásevich,
A homotopic deformation along p of a Leray-Schauder degree result and existence for $(|u'|^{p- 2}u')'+ f(t, u) = 0, u (0) = u (T) = 0, p> 1$, J. Differential Equations, 80 (1989), 1-13.
doi: 10.1016/0022-0396(89)90093-4. |
[13] |
L. Sankar, Classes of Singular Nonlinear Eigenvalue Problems with Semipositone Structure,
Ph. D. thesis, Mississippi State University, 2013. |
[14] |
N. N. Semenov, Chemical Kinetics and Chain Reactions, Oxford University Press, London, 1935.
![]() |
[15] |
R. Shivaji, I. Sim and B. Son,
A uniqueness result for a semipositone p-Laplacian problem on the exterior of a ball, J. Math. Anal. Appl., 445 (2017), 459-475.
doi: 10.1016/j.jmaa.2016.07.029. |
[16] |
Y. B. Zeldovich, G. I. Barenblatt, V. B. Librovich and G. M. Makhviladze, The Mathematical
Theory of Combustion and Explosions, Consultants Bureau, New York, 1985.
doi: 10.1007/978-1-4613-2349-5. |
show all references
References:
[1] |
H. Amann,
Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709.
doi: 10.1137/1018114. |
[2] |
D. Butler, E. Ko, E. K. Lee and R. Shivaji,
Positive radial solutions for elliptic equations on exterior domains with nonlinear boundary conditions, Commun. Pure Appl. Anal., 13 (2014), 2713-2731.
doi: 10.3934/cpaa.2014.13.2713. |
[3] |
R. S. Cantrell and C. Cosner,
Density dependent behavior at habitat boundaries and the allee effect, Bull. Math. Biol., 69 (2007), 2339-2360.
doi: 10.1007/s11538-007-9222-0. |
[4] |
R. S. Cantrell and C. Cosner, Spatial ecology via reaction-diffusion equations, John Wiley &
Sons, Chichester, 2004.
doi: 10.1002/0470871296. |
[5] |
D. Daners,
Robin boundary value problems on arbitrary domains, Trans. Amer. Math. Soc., 352 (2000), 4207-4236.
doi: 10.1090/S0002-9947-00-02444-2. |
[6] |
D. A. Frank-Kamenetskii, Diffusion and Heat Transfer in Chemical Kinetics, New York, Plenum Press, 1969.
![]() |
[7] |
J. Goddard II, E. K. Lee and R. Shivaji,
Population models with diffusion, strong allee effect, and nonlinear boundary conditions, Nonlinear Anal., 74 (2011), 6202-6208.
doi: 10.1016/j.na.2011.06.001. |
[8] |
D. D. Hai,
Uniqueness of positive solutions for a class of quasilinear problems, Nonlinear Anal., 69 (2008), 2720-2732.
doi: 10.1016/j.na.2007.08.046. |
[9] |
E. Ko, M. Ramaswamy and R. Shivaji,
Uniqueness of positive radial solutions for a class of semipositone problems on the exterior of a ball, J. Math. Anal. Appl., 423 (2015), 399-409.
doi: 10.1016/j.jmaa.2014.09.058. |
[10] |
E. K. Lee, R. Shivaji and B. Son,
Positive radial solutions to classes of singular problems on the exterior domain of a ball, J. Math. Anal. Appl., 434 (2016), 1597-1611.
doi: 10.1016/j.jmaa.2015.09.072. |
[11] |
P. Drábek, Topological and Variational Methods for Nonlinear Boundary Value Problems,
1st edition, Addison Wesley Longman Limited, Harlow, 1997 |
[12] |
M. D. Pino, M. Elgueta and R. Manásevich,
A homotopic deformation along p of a Leray-Schauder degree result and existence for $(|u'|^{p- 2}u')'+ f(t, u) = 0, u (0) = u (T) = 0, p> 1$, J. Differential Equations, 80 (1989), 1-13.
doi: 10.1016/0022-0396(89)90093-4. |
[13] |
L. Sankar, Classes of Singular Nonlinear Eigenvalue Problems with Semipositone Structure,
Ph. D. thesis, Mississippi State University, 2013. |
[14] |
N. N. Semenov, Chemical Kinetics and Chain Reactions, Oxford University Press, London, 1935.
![]() |
[15] |
R. Shivaji, I. Sim and B. Son,
A uniqueness result for a semipositone p-Laplacian problem on the exterior of a ball, J. Math. Anal. Appl., 445 (2017), 459-475.
doi: 10.1016/j.jmaa.2016.07.029. |
[16] |
Y. B. Zeldovich, G. I. Barenblatt, V. B. Librovich and G. M. Makhviladze, The Mathematical
Theory of Combustion and Explosions, Consultants Bureau, New York, 1985.
doi: 10.1007/978-1-4613-2349-5. |
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