Analysis of positive solutions for a class of semipositone <i>p</i>-Laplacian problems with nonlinear boundary conditions

Eun Kyoung Lee; R. Shivaji; Inbo Sim; Byungjae Son

doi:10.3934/cpaa.2019055

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May 2019, 18(3): 1139-1154. doi: 10.3934/cpaa.2019055

Analysis of positive solutions for a class of semipositone p-Laplacian problems with nonlinear boundary conditions

Eun Kyoung Lee ^1, , R. Shivaji ^2, , Inbo Sim ^3,, and Byungjae Son ^4,

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

* Corresponding author: Inbo Sim

Received February 2018 Revised August 2018 Published November 2018

Fund Project: The third author is supported by the National Research Foundation of Korea Grant funded by the Korea Government (MEST) (NRF-2018R1D1A3A03000678).

We study positive solutions to (singular) boundary value problems of the form:

$\left\{ \begin{align} & -\left( {{\varphi }_{p}}(u') \right)'=\lambda h(t)\frac{f(u)}{{{u}^{\alpha }}},~\ \ t\in (0,1),~~ \\ & u'(1)+c(u(1))u(1)=0,~ \\ & u(0)=0, \\ \end{align} \right.$

where

$\varphi_p(u): = |u|^{p-2}u$

with

$p>1$

is the

$p$

-Laplacian operator of

$u$

$λ>0$

$0≤α<1$

$c:[0,∞)\rightarrow (0,∞)$

is continuous and

$h:(0,1)\rightarrow (0,∞)$

is continuous and integrable. We assume that

$f∈ C[0,∞)$

is such that

$f(0)<0$

$\lim_{s\rightarrow ∞}f(s) = ∞$

and

$\frac{f(s)}{s^{α}}$

has a

$p$

-sublinear growth at infinity, namely,

$\lim_{s \rightarrow ∞}\frac{f(s)}{s^{p-1+α}} = 0$

. We will discuss nonexistence results for

$λ≈ 0$

, and existence and uniqueness results for

$λ \gg 1$

. We establish the existence result by a method of sub-supersolutions and the uniqueness result by establishing growth estimates for solutions.

Keywords: Semipositone, p-Laplacian, p-sublinear growth at infinity, nonlinear boundary conditions, positive solutions, existence, uniqueness.

Mathematics Subject Classification: Primary: 34B16, 34B18; Secondary: 35J57.

Citation: Eun Kyoung Lee, R. Shivaji, Inbo Sim, Byungjae Son. Analysis of positive solutions for a class of semipositone p-Laplacian problems with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1139-1154. doi: 10.3934/cpaa.2019055

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[4]	R. S. Cantrell and C. Cosner, Spatial ecology via reaction-diffusion equations, John Wiley & Sons, Chichester, 2004. doi: 10.1002/0470871296. Google Scholar
[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. Google Scholar
[6]	D. A. Frank-Kamenetskii, Diffusion and Heat Transfer in Chemical Kinetics, New York, Plenum Press, 1969. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[11]	P. Drábek, Topological and Variational Methods for Nonlinear Boundary Value Problems, 1^st edition, Addison Wesley Longman Limited, Harlow, 1997 Google Scholar
[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. Google Scholar
[13]	L. Sankar, Classes of Singular Nonlinear Eigenvalue Problems with Semipositone Structure, Ph. D. thesis, Mississippi State University, 2013. Google Scholar
[14]	N. N. Semenov, Chemical Kinetics and Chain Reactions, Oxford University Press, London, 1935. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[4]	R. S. Cantrell and C. Cosner, Spatial ecology via reaction-diffusion equations, John Wiley & Sons, Chichester, 2004. doi: 10.1002/0470871296. Google Scholar
[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. Google Scholar
[6]	D. A. Frank-Kamenetskii, Diffusion and Heat Transfer in Chemical Kinetics, New York, Plenum Press, 1969. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[11]	P. Drábek, Topological and Variational Methods for Nonlinear Boundary Value Problems, 1^st edition, Addison Wesley Longman Limited, Harlow, 1997 Google Scholar
[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. Google Scholar
[13]	L. Sankar, Classes of Singular Nonlinear Eigenvalue Problems with Semipositone Structure, Ph. D. thesis, Mississippi State University, 2013. Google Scholar
[14]	N. N. Semenov, Chemical Kinetics and Chain Reactions, Oxford University Press, London, 1935. Google Scholar
[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. Google Scholar
[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. Google Scholar

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Analysis of positive solutions for a class of semipositone p-Laplacian problems with nonlinear boundary conditions

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