• 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. ButlerE. KoE. 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 IIE. 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. KoM. 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. LeeR. 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, 1997 Google Scholar

[12]

M. D. PinoM. 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. ShivajiI. 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. ButlerE. KoE. 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 IIE. 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. KoM. 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. LeeR. 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, 1997 Google Scholar

[12]

M. D. PinoM. 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. ShivajiI. 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]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[2]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442

[3]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[4]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445

[5]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[6]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[7]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461

[8]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020267

[9]

Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020378

[10]

Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383

[11]

Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252

[12]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119

[13]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272

[14]

Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020052

[15]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

[16]

Shiqiu Fu, Kanishka Perera. On a class of semipositone problems with singular Trudinger-Moser nonlinearities. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020452

[17]

Maoding Zhen, Binlin Zhang, Vicenţiu D. Rădulescu. Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020379

[18]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276

[19]

Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456

[20]

Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (164)
  • HTML views (233)
  • Cited by (2)

Other articles
by authors

[Back to Top]