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

Opuscula Mathematica

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.

Full text (pdf)

  1. R. Agarwal, D. O'Regan, Semipositone Dirichlet boundary value problems with singular nonlinearities, Houston J. Math. 30 (2004), 297-308.
  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. W. Allegretto, P. Nistri, P. Zecca, Positive solutions for elliptic nonpositone problems, Differential Integral Equations 5 (1992), 95-101.
  4. H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach Spaces, SIAM Rev. 18 (1976), 620-709.
  5. A. Ambrosetti, D. Arcoya, B. Buffoni, Positive solutions for some semipositone problems via bifurcation theory, Differential Integral Equations 7 (1994), 655-663.
  6. V. Anurada, D.D. Hai, R. Shivaji, Existence results for superlinear semipositone BVP's, Proc. Amer. Math. Soc. 124 (1996), 757-763.
  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. L. Erbe, H. Wang, On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc. 120 (1994) 3, 743-748.
  9. D.D. Hai, On singular Sturm-Liouville boundary-value problems, Proc. Roy. Soc. Edinburgh Sect. A 140 (2010) 1, 49-63.
  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. J. Jacobsen, K. Schmitt, Radial solutions of quasilinear elliptic differential equations, Handbook of Differential Equations, vol. 1, North-Holland, 2004, 359-435.
  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. 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. Y. Liu, Twin solutions to singular semipositone problems, J. Math. Anal. Appl. 286 (2003), 248-260.
  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. 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. J. Wang, The existence of positive solutions for the one-dimensional \(p\)-Laplacian, Proc. Amer. Math. Soc. 125 (1997), 2275-2283.
  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. Q. Yao, An existence theorem of a positive solution to a semipositone Sturm-Liouville boundary value problem, Appl. Math. Lett. 23 (2010), 1401-1406.
  • Communicated by Jean Mawhin.
  • Received: 2019-02-18.
  • Accepted: 2019-05-26.
  • Published online: 2019-09-05.
Opuscula Mathematica - cover

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

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.