# American Institute of Mathematical Sciences

• Previous Article
Backward compact and periodic random attractors for non-autonomous sine-Gordon equations with multiplicative noise
• CPAA Home
• This Issue
• Next Article
The exponential behavior of a stochastic Cahn-Hilliard-Navier-Stokes model with multiplicative noise
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. 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, 1st edition, Addison Wesley Longman Limited, Harlow, 1997Google 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, 1st edition, Addison Wesley Longman Limited, Harlow, 1997Google 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
 [1] Maya Chhetri, D. D. Hai, R. Shivaji. On positive solutions for classes of p-Laplacian semipositone systems. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 1063-1071. doi: 10.3934/dcds.2003.9.1063 [2] Yinbin Deng, Yi Li, Wei Shuai. Existence of solutions for a class of p-Laplacian type equation with critical growth and potential vanishing at infinity. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 683-699. doi: 10.3934/dcds.2016.36.683 [3] Michael E. Filippakis, Nikolaos S. Papageorgiou. Existence and multiplicity of positive solutions for nonlinear boundary value problems driven by the scalar $p$-Laplacian. Communications on Pure & Applied Analysis, 2004, 3 (4) : 729-756. doi: 10.3934/cpaa.2004.3.729 [4] Leszek Gasiński. Positive solutions for resonant boundary value problems with the scalar p-Laplacian and nonsmooth potential. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 143-158. doi: 10.3934/dcds.2007.17.143 [5] Zuodong Yang, Jing Mo, Subei Li. Positive solutions of $p$-Laplacian equations with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 623-636. doi: 10.3934/dcdsb.2011.16.623 [6] Petru Jebelean. Infinitely many solutions for ordinary $p$-Laplacian systems with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2008, 7 (2) : 267-275. doi: 10.3934/cpaa.2008.7.267 [7] Sophia Th. Kyritsi, Nikolaos S. Papageorgiou. Positive solutions for p-Laplacian equations with concave terms. Conference Publications, 2011, 2011 (Special) : 922-930. doi: 10.3934/proc.2011.2011.922 [8] Leyun Wu, Pengcheng Niu. Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1573-1583. doi: 10.3934/dcds.2019069 [9] Friedemann Brock, Leonelo Iturriaga, Justino Sánchez, Pedro Ubilla. Existence of positive solutions for $p$--Laplacian problems with weights. Communications on Pure & Applied Analysis, 2006, 5 (4) : 941-952. doi: 10.3934/cpaa.2006.5.941 [10] John R. Graef, Lingju Kong. Uniqueness and parameter dependence of positive solutions of third order boundary value problems with $p$-laplacian. Conference Publications, 2011, 2011 (Special) : 515-522. doi: 10.3934/proc.2011.2011.515 [11] Dimitri Mugnai. Bounce on a p-Laplacian. Communications on Pure & Applied Analysis, 2003, 2 (3) : 371-379. doi: 10.3934/cpaa.2003.2.371 [12] E. N. Dancer, Zhitao Zhang. Critical point, anti-maximum principle and semipositone p-laplacian problems. Conference Publications, 2005, 2005 (Special) : 209-215. doi: 10.3934/proc.2005.2005.209 [13] Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1801-1816. doi: 10.3934/dcdsb.2017107 [14] Zhong Tan, Zheng-An Yao. The existence and asymptotic behavior of the evolution p-Laplacian equations with strong nonlinear sources. Communications on Pure & Applied Analysis, 2004, 3 (3) : 475-490. doi: 10.3934/cpaa.2004.3.475 [15] Shanming Ji, Jingxue Yin, Yutian Li. Positive periodic solutions of the weighted $p$-Laplacian with nonlinear sources. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2411-2439. doi: 10.3934/dcds.2018100 [16] Francisco Odair de Paiva, Humberto Ramos Quoirin. Resonance and nonresonance for p-Laplacian problems with weighted eigenvalues conditions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1219-1227. doi: 10.3934/dcds.2009.25.1219 [17] Ronghua Jiang, Jun Zhou. Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1205-1226. doi: 10.3934/cpaa.2019058 [18] CÉSAR E. TORRES LEDESMA. Existence and symmetry result for fractional p-Laplacian in $\mathbb{R}^{n}$. Communications on Pure & Applied Analysis, 2017, 16 (1) : 99-114. doi: 10.3934/cpaa.2017004 [19] Lingyu Jin, Yan Li. A Hopf's lemma and the boundary regularity for the fractional p-Laplacian. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1477-1495. doi: 10.3934/dcds.2019063 [20] Bernd Kawohl, Jiří Horák. On the geometry of the p-Laplacian operator. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 799-813. doi: 10.3934/dcdss.2017040

2018 Impact Factor: 0.925