Opuscula Mathematica
Opuscula Math. 39, no. 5 (2019), 675-689

Positive solutions for the one-dimensional p-Laplacian with nonlinear boundary conditions

D. D. Hai
X. Wang
Abstract. We prove the existence of positive solutions for the \(p\)-Laplacian problem \[\begin{cases}-(r(t)\phi (u^{\prime }))^{\prime }=\lambda g(t)f(u),& t\in (0,1),\\au(0)-H_{1}(u^{\prime }(0))=0,\\cu(1)+H_{2}(u^{\prime}(1))=0,\end{cases}\] where \(\phi (s)=|s|^{p-2}s\), \(p\gt 1\), \(H_{i}:\mathbb{R}\rightarrow\mathbb{R}\) can be nonlinear, \(i=1,2\), \(f:(0,\infty )\rightarrow \mathbb{R}\) is \(p\)-superlinear or \(p\)-sublinear at \(\infty\) and is allowed be singular \((\pm\infty)\) at \(0\), and \(\lambda\) is a positive parameter.
Keywords: \(p\)-Laplacian, semipositone, nonlinear boundary conditions, positive solutions.
Mathematics Subject Classification: 34B16, 34B18.
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References ▼
  1. [1] R. Agarwal, D. O'Regan, Semipositone Dirichlet boundary value problems with singular nonlinearities, Houston J. Math. 30 (2004), 297-308.
  2. [2] R. Agarwal, D. Cao, H. Lu, Existence and multiplicity of positive solutions for singular semipositone \(p\)-Laplacian equations, Can. J. Math. 58 (2006), 449-475.
  3. [3] W. Allegretto, P. Nistri, P. Zecca, Positive solutions for elliptic nonpositone problems, Differential Integral Equations 5 (1992), 95-101.
  4. [4] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach Spaces, SIAM Rev. 18 (1976), 620-709.
  5. [5] A. Ambrosetti, D. Arcoya, B. Buffoni, Positive solutions for some semipositone problems via bifurcation theory, Differential Integral Equations 7 (1994), 655-663.
  6. [6] V. Anurada, D.D. Hai, R. Shivaji, Existence results for superlinear semipositone BVP's, Proc. Amer. Math. Soc. 124 (1996), 757-763.
  7. [7] D. Arcoya, A. Zertiti, Existence and nonexistence of radially symmetric nonnegative solutions for a class of semipositone problems in an annulus, Rend. Mat. 14 (1994), 625-646.
  8. [8] L. Erbe, H. Wang, On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc. 120 (1994) 3, 743-748.
  9. [9] D.D. Hai, On singular Sturm-Liouville boundary-value problems, Proc. Roy. Soc. Edinburgh Sect. A 140 (2010) 1, 49-63.
  10. [10] D.D. Hai, Existence of positive solutions for singular \(p\)-Laplacian Sturm-Liouville boundary value problems, Electron. J. Differential Equations (2016), paper no. 260.
  11. [11] J. Jacobsen, K. Schmitt, Radial solutions of quasilinear elliptic differential equations, Handbook of Differential Equations, vol. 1, North-Holland, 2004, 359-435.
  12. [12] K. Lan, X. Yang, G. Yang, Positive solutions of one-dimensional \(p\)-Laplacian equations and applications to population models of one species, Topol. Methods Nonlinear Anal. 46 (2015), 431-445.
  13. [13] E. Lee, R. Shivaji, J. Ye, Subsolutions: A journey from positone to infinite semipositone problems, Electron. J. Differ. Equ. Conf. 17 (2009), 123-131.
  14. [14] Y. Liu, Twin solutions to singular semipositone problems, J. Math. Anal. Appl. 286 (2003), 248-260.
  15. [15] J. Smoller, A. Wasserman, Existence of positive solutions for semilinear elliptic equations in general domains, Arch. Ration. Mech. Anal. 98 (1987), 229-249.
  16. [16] J.R.L. Webb, K.Q. Lan, Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary vale problems of local and nonlocal types, Topol. Methods Nonlinear Anal. 27 (2006), 91-116.
  17. [17] J. Wang, The existence of positive solutions for the one-dimensional \(p\)-Laplacian, Proc. Amer. Math. Soc. 125 (1997), 2275-2283.
  18. [18] G.C. Yang, P.F. Zhou, A new existence results of positive solutions for the Sturm-Liouville boundary value problem, Appl. Math. Lett. 23 (2010), 1401-1406.
  19. [19] Q. Yao, An existence theorem of a positive solution to a semipositone Sturm-Liouville boundary value problem, Appl. Math. Lett. 23 (2010), 1401-1406.
About the authors ▼
D. D. Hai
https://orcid.org/0000-0002-9927-0793
Mississippi State University, Department of Mathematics and Statistics, Mississippi State, MS 39762, USA
X. Wang
https://orcid.org/0000-0001-5584-5009
Mississippi State University, Department of Mathematics and Statistics, Mississippi State, MS 39762, USA
About the article ▼
Communicated by Jean Mawhin.
Received: 2019-02-18.
Accepted: 2019-05-26.
Published online: 2019-09-05.
Cite this article as ▼
D. D. Hai, X. Wang, Positive solutions for the one-dimensional p-Laplacian with nonlinear boundary conditions, Opuscula Math. 39, no. 5 (2019), 675-689, https://doi.org/10.7494/OpMath.2019.39.5.675.
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