Opuscula Math. 39, no. 5 (2019), 675-689
https://doi.org/10.7494/OpMath.2019.39.5.675

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.

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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.
• Accepted: 2019-05-26.
• Published online: 2019-09-05.