    • Previous Article
Uniqueness and stability of time-periodic pyramidal fronts for a periodic competition-diffusion system
• CPAA Home
• This Issue
• Next Article
Multiplicity of positive solutions to nonlinear systems of Hammerstein integral equations with weighted functions
January  2020, 19(1): 241-252. doi: 10.3934/cpaa.2020013

## Positive solutions for the one-dimensional singular superlinear $p$-Laplacian problem

 1 Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam 2 Department of Mathematics and Statistics, Mississippi state University, Mississippi State, MS 39762, USA

*Corresponding author

Received  November 2018 Revised  February 2019 Published  July 2019

We prove the existence of positive classical solutions for the
 $p$
-Laplacian problem
 $\begin{equation*} \left\{ \begin{array}{c} -(r(t)\phi (u^{\prime }))^{\prime } = -\frac{\lambda }{u^{\delta }}+f(t,u),\ t\in (0,1), \\ u(0) = u(1) = 0,\end{array}\right. \end{equation*}$
where
 $0<\delta <1$
,
 $\phi (s) = |s|^{p-2}s$
,
 $p>1$
,
 $f:(0,1)\times \lbrack 0,\infty )\rightarrow \mathbb{R}$
is a Carathéodory function satisfying
 $\limsup\limits_{z\rightarrow 0^{+}}\frac{f(t,z)}{z^{p-1}}<\lambda _{1}<\liminf\limits_{z\rightarrow \infty }\frac{f(t,z)}{z^{p-1}}$
uniformly for a.e.
 $t$
 $\in (0,1),$
where
 $\lambda_{1}$
denotes the principal eigenvalue of
 $-(r(t)\phi (u^{\prime }))^{\prime }$
with zero boundary conditions, and
 $\lambda$
is a small nonnegative parameter.
Citation: K. D. Chu, D. D. Hai. Positive solutions for the one-dimensional singular superlinear $p$-Laplacian problem. Communications on Pure & Applied Analysis, 2020, 19 (1) : 241-252. doi: 10.3934/cpaa.2020013
##### 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  H. Brezis, Analyse Fonctionnnelle, Theorie et Applications, in French, Second Edition, Paris: Masson, 1983. Google Scholar  A. Castro, C. Maya and R. Shivaji, Nonlinear eigenvalue problems with semipositone structure, Proceedings of the Conference on Nonlinear Differential Equations (Coral Gables, FL, 1999), 33–49, Electron. J. Differ. Equ. Conf., 5, Southwest Texas State Univ., San Marcos, TX, 2000. Google Scholar  P. Candito, S. Carl and R. Livrea, Multiple solutions for quasilinear elliptic problems via critical points in open sublevels and truncation principles, J. Math. Anal. Appl., 395 (2012), 156–163. doi: 10.1016/j.jmaa.2012.05.003.  Google Scholar  P. Candito, S. Carl and R. Livrea, Variational versus pseudomonotone operator approach in parameter-dependent nonlinear elliptic problems, Dynam. Systems Appl., 22 (2013), 397–410. Google Scholar  P. Candito, P, S. Carl and R. Livrea, Critical points in open sublevels and multiple solutions for parameter-depending quasilinear elliptic equations, Adv. Differential Equations, 19 (2014), 1021–1042. Google Scholar  K. D. Chu and D. D. Hai, Positive solutions for the one-dimensional Sturm-Liouville superlinear $p$-Laplacian problem, Electron. J. Differential Equations, 92 (2018), 1–14. Google Scholar  H. Dang, K. Schmitt and R. Shivaji, On the number of solutions of boundary value problems involving the p-Laplacian, Electron. J. Differential Equations, 1 (1996), 1–9. Google Scholar  A. Cwiszewski and M. Maciejewski, Positive stationary solutions for $p$-Laplacian problems with nonpositive perturbation, J. Differential Equations, 254 (2013), 1120–1136. doi: 10.1016/j.jde.2012.10.004.  Google Scholar  C. De Coster, Pairs of positive solutions for the one-dimensional $p$-Laplacian, Nonlinear Anal., 23 (1994), 669–681. doi: 10.1016/0362-546X(94)90245-3.  Google Scholar  M. Del Pino, M. Elgueta and R. Manasevich, A homotopic deformation along $p$ of a Leray-Schauder degree result and existence for $(|u^{\prime }|^{p-2}u^{\prime })^{\prime }+f(t, u) = 0, u(0) = u(T) = 0, p>1$, J. Differential Equations, 80 (1989), 1–13. doi: 10.1016/0022-0396(89)90093-4.  Google Scholar  P. Drabek, Ranges of a -homogeneous operators and their perturbations, Casopis Pest. Mat., 105 (1980), 167–183. Google Scholar  L. Erbe and H. Wang, On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc., 120 (1994), 743–748. doi: 10.2307/2160465.  Google Scholar  P. L. Lions and R. D. Nusbaum, Estimations a priori pour les solutions positives de problèmes elliptiques superlin éaires, C. R. Acad. Sci. Paris Sér. A-B, 290 (1980), 217–220. Google Scholar  D. D. Hai, On singular Sturm-Liouville boundary-value problems, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 49–63. doi: 10.1017/S0308210508000358.  Google Scholar  H. G. Kaper, M. Knaap and M. K. Kwong, Existence theorems for second order boundary value problems, Differential Integral Equations, 4 (1991), 543–554. Google Scholar  L. Kong and Q. Kong, Right-indefinite half-linear Sturm-Liouville problems, Comput. Math. Appl., 55 (2008), 2554–2564. doi: 10.1016/j.camwa.2007.10.008.  Google Scholar  E. Lee, R. Shivaji and J. Ye, Subsolutions: a journey from positone to infinite semipositone problems, Electron. J. Differ. Equ. Conf., 17 (2009), 123–131. Google Scholar  R. Manásevich, F. Njoku and F. Zanolin, Positive solutions for the one-dimensional $p$-Laplacian, Differential Integral Equations, 8 (1995), 213–222. Google Scholar  T. Oden, Qualitative Methods in Nonlinear Mechanics, Englewood Cliffs, NJ, 1986. Google Scholar  S. Sakaguchi, Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems, Ann. Scuola Norm. Sup. pisa Cl. Sci, 14 (1987), 403–421. 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  H. Brezis, Analyse Fonctionnnelle, Theorie et Applications, in French, Second Edition, Paris: Masson, 1983. Google Scholar  A. Castro, C. Maya and R. Shivaji, Nonlinear eigenvalue problems with semipositone structure, Proceedings of the Conference on Nonlinear Differential Equations (Coral Gables, FL, 1999), 33–49, Electron. J. Differ. Equ. Conf., 5, Southwest Texas State Univ., San Marcos, TX, 2000. Google Scholar  P. Candito, S. Carl and R. Livrea, Multiple solutions for quasilinear elliptic problems via critical points in open sublevels and truncation principles, J. Math. Anal. Appl., 395 (2012), 156–163. doi: 10.1016/j.jmaa.2012.05.003.  Google Scholar  P. Candito, S. Carl and R. Livrea, Variational versus pseudomonotone operator approach in parameter-dependent nonlinear elliptic problems, Dynam. Systems Appl., 22 (2013), 397–410. Google Scholar  P. Candito, P, S. Carl and R. Livrea, Critical points in open sublevels and multiple solutions for parameter-depending quasilinear elliptic equations, Adv. Differential Equations, 19 (2014), 1021–1042. Google Scholar  K. D. Chu and D. D. Hai, Positive solutions for the one-dimensional Sturm-Liouville superlinear $p$-Laplacian problem, Electron. J. Differential Equations, 92 (2018), 1–14. Google Scholar  H. Dang, K. Schmitt and R. Shivaji, On the number of solutions of boundary value problems involving the p-Laplacian, Electron. J. Differential Equations, 1 (1996), 1–9. Google Scholar  A. Cwiszewski and M. Maciejewski, Positive stationary solutions for $p$-Laplacian problems with nonpositive perturbation, J. Differential Equations, 254 (2013), 1120–1136. doi: 10.1016/j.jde.2012.10.004.  Google Scholar  C. De Coster, Pairs of positive solutions for the one-dimensional $p$-Laplacian, Nonlinear Anal., 23 (1994), 669–681. doi: 10.1016/0362-546X(94)90245-3.  Google Scholar  M. Del Pino, M. Elgueta and R. Manasevich, A homotopic deformation along $p$ of a Leray-Schauder degree result and existence for $(|u^{\prime }|^{p-2}u^{\prime })^{\prime }+f(t, u) = 0, u(0) = u(T) = 0, p>1$, J. Differential Equations, 80 (1989), 1–13. doi: 10.1016/0022-0396(89)90093-4.  Google Scholar  P. Drabek, Ranges of a -homogeneous operators and their perturbations, Casopis Pest. Mat., 105 (1980), 167–183. Google Scholar  L. Erbe and H. Wang, On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc., 120 (1994), 743–748. doi: 10.2307/2160465.  Google Scholar  P. L. Lions and R. D. Nusbaum, Estimations a priori pour les solutions positives de problèmes elliptiques superlin éaires, C. R. Acad. Sci. Paris Sér. A-B, 290 (1980), 217–220. Google Scholar  D. D. Hai, On singular Sturm-Liouville boundary-value problems, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 49–63. doi: 10.1017/S0308210508000358.  Google Scholar  H. G. Kaper, M. Knaap and M. K. Kwong, Existence theorems for second order boundary value problems, Differential Integral Equations, 4 (1991), 543–554. Google Scholar  L. Kong and Q. Kong, Right-indefinite half-linear Sturm-Liouville problems, Comput. Math. Appl., 55 (2008), 2554–2564. doi: 10.1016/j.camwa.2007.10.008.  Google Scholar  E. Lee, R. Shivaji and J. Ye, Subsolutions: a journey from positone to infinite semipositone problems, Electron. J. Differ. Equ. Conf., 17 (2009), 123–131. Google Scholar  R. Manásevich, F. Njoku and F. Zanolin, Positive solutions for the one-dimensional $p$-Laplacian, Differential Integral Equations, 8 (1995), 213–222. Google Scholar  T. Oden, Qualitative Methods in Nonlinear Mechanics, Englewood Cliffs, NJ, 1986. Google Scholar  S. Sakaguchi, Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems, Ann. Scuola Norm. Sup. pisa Cl. Sci, 14 (1987), 403–421. Google Scholar
  Shanming Ji, Yutian Li, Rui Huang, Xuejing Yin. Singular periodic solutions for the p-laplacian ina punctured domain. Communications on Pure & Applied Analysis, 2017, 16 (2) : 373-392. doi: 10.3934/cpaa.2017019  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  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  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  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  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  Dimitri Mugnai. Bounce on a p-Laplacian. Communications on Pure & Applied Analysis, 2003, 2 (3) : 371-379. doi: 10.3934/cpaa.2003.2.371  Leszek Gasiński, Nikolaos S. Papageorgiou. Three nontrivial solutions for periodic problems with the $p$-Laplacian and a $p$-superlinear nonlinearity. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1421-1437. doi: 10.3934/cpaa.2009.8.1421  Robert Stegliński. On homoclinic solutions for a second order difference equation with p-Laplacian. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 487-492. doi: 10.3934/dcdsb.2018033  Everaldo S. de Medeiros, Jianfu Yang. Asymptotic behavior of solutions to a perturbed p-Laplacian problem with Neumann condition. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 595-606. doi: 10.3934/dcds.2005.12.595  Adam Lipowski, Bogdan Przeradzki, Katarzyna Szymańska-Dębowska. Periodic solutions to differential equations with a generalized p-Laplacian. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2593-2601. doi: 10.3934/dcdsb.2014.19.2593  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  M. Gaudenzi, P. Habets, F. Zanolin. Positive solutions of superlinear boundary value problems with singular indefinite weight. Communications on Pure & Applied Analysis, 2003, 2 (3) : 411-423. doi: 10.3934/cpaa.2003.2.411  Nikolaos S. Papageorgiou, George Smyrlis. Positive solutions for parametric $p$-Laplacian equations. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1545-1570. doi: 10.3934/cpaa.2016002  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  Yansheng Zhong, Yongqing Li. On a p-Laplacian eigenvalue problem with supercritical exponent. Communications on Pure & Applied Analysis, 2019, 18 (1) : 227-236. doi: 10.3934/cpaa.2019012  Genni Fragnelli, Dimitri Mugnai, Nikolaos S. Papageorgiou. Robin problems for the p-Laplacian with gradient dependence. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 287-295. doi: 10.3934/dcdss.2019020  Francesca Colasuonno, Benedetta Noris. A p-Laplacian supercritical Neumann problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3025-3057. doi: 10.3934/dcds.2017130  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  Nikolaos S. Papageorgiou, Vicenţiu D. Rǎdulescu, Dušan D. Repovš. Nodal solutions for the Robin p-Laplacian plus an indefinite potential and a general reaction term. Communications on Pure & Applied Analysis, 2018, 17 (1) : 231-241. doi: 10.3934/cpaa.2018014

2018 Impact Factor: 0.925