    • 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:
  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  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  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  R. S. Cantrell and C. Cosner, Spatial ecology via reaction-diffusion equations, John Wiley & Sons, Chichester, 2004. doi: 10.1002/0470871296.  Google Scholar  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  D. A. Frank-Kamenetskii, Diffusion and Heat Transfer in Chemical Kinetics, New York, Plenum Press, 1969.   Google Scholar  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  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  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  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  P. Drábek, Topological and Variational Methods for Nonlinear Boundary Value Problems, 1st edition, Addison Wesley Longman Limited, Harlow, 1997 Google Scholar  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  L. Sankar, Classes of Singular Nonlinear Eigenvalue Problems with Semipositone Structure, Ph. D. thesis, Mississippi State University, 2013. Google Scholar  N. N. Semenov, Chemical Kinetics and Chain Reactions, Oxford University Press, London, 1935.   Google Scholar  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  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:
  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  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  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  R. S. Cantrell and C. Cosner, Spatial ecology via reaction-diffusion equations, John Wiley & Sons, Chichester, 2004. doi: 10.1002/0470871296.  Google Scholar  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  D. A. Frank-Kamenetskii, Diffusion and Heat Transfer in Chemical Kinetics, New York, Plenum Press, 1969.   Google Scholar  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  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  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  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  P. Drábek, Topological and Variational Methods for Nonlinear Boundary Value Problems, 1st edition, Addison Wesley Longman Limited, Harlow, 1997 Google Scholar  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  L. Sankar, Classes of Singular Nonlinear Eigenvalue Problems with Semipositone Structure, Ph. D. thesis, Mississippi State University, 2013. Google Scholar  N. N. Semenov, Chemical Kinetics and Chain Reactions, Oxford University Press, London, 1935.   Google Scholar  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  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
  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  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  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  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  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  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  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  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  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  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  Dimitri Mugnai. Bounce on a p-Laplacian. Communications on Pure & Applied Analysis, 2003, 2 (3) : 371-379. doi: 10.3934/cpaa.2003.2.371  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  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  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  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  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  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  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  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  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