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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

 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.
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. [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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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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