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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:
 [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] H. Brezis, Analyse Fonctionnnelle, Theorie et Applications, in French, Second Edition, Paris: Masson, 1983.  Google Scholar [3] 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 [4] 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 [5] 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 [6] 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 [7] 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 [8] 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 [9] 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 [10] 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 [11] 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 [12] P. Drabek, Ranges of a -homogeneous operators and their perturbations, Casopis Pest. Mat., 105 (1980), 167–183.  Google Scholar [13] 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 [14] 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 [15] 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 [16] 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 [17] 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 [18] 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 [19] R. Manásevich, F. Njoku and F. Zanolin, Positive solutions for the one-dimensional $p$-Laplacian, Differential Integral Equations, 8 (1995), 213–222.  Google Scholar [20] T. Oden, Qualitative Methods in Nonlinear Mechanics, Englewood Cliffs, NJ, 1986. Google Scholar [21] 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

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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.  Google Scholar [2] H. Brezis, Analyse Fonctionnnelle, Theorie et Applications, in French, Second Edition, Paris: Masson, 1983.  Google Scholar [3] 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 [4] 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 [5] 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 [6] 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 [7] 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 [8] 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 [9] 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 [10] 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 [11] 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 [12] P. Drabek, Ranges of a -homogeneous operators and their perturbations, Casopis Pest. Mat., 105 (1980), 167–183.  Google Scholar [13] 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 [14] 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 [15] 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 [16] 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 [17] 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 [18] 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 [19] R. Manásevich, F. Njoku and F. Zanolin, Positive solutions for the one-dimensional $p$-Laplacian, Differential Integral Equations, 8 (1995), 213–222.  Google Scholar [20] T. Oden, Qualitative Methods in Nonlinear Mechanics, Englewood Cliffs, NJ, 1986. Google Scholar [21] 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
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