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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 and 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.

[2]

H. Brezis, Analyse Fonctionnnelle, Theorie et Applications, in French, Second Edition, Paris: Masson, 1983.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

P. Drabek, Ranges of a -homogeneous operators and their perturbations, Casopis Pest. Mat., 105 (1980), 167–183.

[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.

[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.

[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.

[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.

[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.

[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.

[19]

R. Manásevich, F. Njoku and F. Zanolin, Positive solutions for the one-dimensional $p$-Laplacian, Differential Integral Equations, 8 (1995), 213–222.

[20]

T. Oden, Qualitative Methods in Nonlinear Mechanics, Englewood Cliffs, NJ, 1986.

[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.

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.

[2]

H. Brezis, Analyse Fonctionnnelle, Theorie et Applications, in French, Second Edition, Paris: Masson, 1983.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

P. Drabek, Ranges of a -homogeneous operators and their perturbations, Casopis Pest. Mat., 105 (1980), 167–183.

[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.

[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.

[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.

[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.

[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.

[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.

[19]

R. Manásevich, F. Njoku and F. Zanolin, Positive solutions for the one-dimensional $p$-Laplacian, Differential Integral Equations, 8 (1995), 213–222.

[20]

T. Oden, Qualitative Methods in Nonlinear Mechanics, Englewood Cliffs, NJ, 1986.

[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.

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