Acta mathematica scientia, Series B >

2017 , Vol. 37 >Issue 5: 1472 - 1482

DOI: https://doi.org/10.1016/S0252-9602(17)30085-1

Articles

MULTIPLE POSITIVE SOLUTIONS FOR A CLASS OF SEMIPOSITONE NEUMANN PROBLEMS WITH SINGULAR φ-LAPLACIAN

  • Ruyun MA ,
  • Hongliang GAO
Expand
  • Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Received date: 2015-11-01

  Revised date: 2017-05-01

  Online published: 2017-10-25

Supported by

Supported by the NSFC (11361054, 11671322)

Fold

Abstract

We study the existence of multiple positive solutions for a Neumann problem with singular φ-Laplacian

where λ is a positive parameter,φ(s)=(s)/√1-s2,f ∈ C1([0,∞),R),f'(u)> 0 for u > 0,and for some 0 < β < θ such that f(u)< 0 for u ∈[0,β)(semipositone) and f(u)> 0 for u > β. Under some suitable assumptions,we obtain the existence of multiple positive solutions of the above problem by using the quadrature technique.Further,if f ∈ C2([0,β)∪(β,∞),R), f"(u) ≥ 0 for u ∈[0,β) and f"(u) ≤ 0 for u ∈(β,∞),then there exist exactly 2n+1 positive solutions for some interval of λ,which is dependent on n and θ.Moreover,We also give some examples to apply our results.

Key words： multiple positive solutions; Neumann problem; prescribed mean curvature equation; time map

Cite this article

Ruyun MA , Hongliang GAO . MULTIPLE POSITIVE SOLUTIONS FOR A CLASS OF SEMIPOSITONE NEUMANN PROBLEMS WITH SINGULAR φ-LAPLACIAN[J]. Acta mathematica scientia, Series B, 2017 , 37(5) : 1472 -1482 . DOI: 10.1016/S0252-9602(17)30085-1

References

[1] Bereanu C, Jebelean P, Mawhin J. Radial solutions for Neumann problems involving mean curvature operators in Euclidean and Minkowsi spaces. Math Nachr, 2010, 283(3):379-391
[2] Bereanu C, Jebelean P, Mawhin J. Radial solutions for Neumann problems with φ-Laplacians and pendulum-like nonlinearities. Discret Cont Dynam Syst A, 2010, 28(2):637-648
[3] Bereanu C, Jebelean P, Torres P J. Multiple solutions for Neumann and periodic problems with singular φ-Laplacian. J Funct Anal, 2011, 261(11):3226-3246
[4] Bereanu C, Jebelean P, Torres P J. Multiple positive radial solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space. J Funct Anal, 2013, 265(4):644-659
[5] Bereanu C, Jebelean P, Torres P J. Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space. J Funct Anal, 2013, 264(1):270-287
[6] Bereanu C, Mawhin J. Nonlinear Neumann boundary value problems with φ-Laplacian operators. An Stiint Univ Ovidius Constanta, 2004, 12(2):73-92
[7] Bereanu C, Mawhin J. Existence and multiplicity results for some nonlinear problems with singular φ-Laplacian. J Differ Equ, 2007, 243(2):536-557
[8] Bonheure D, Noris B, Weth T. Increasing radial solutions for Neumann problems without growth restrictions. Ann Inst H Poincaré Anal Non Linéaire, 2012, 29(4):573-588
[9] Chu J F, Sun Y G, Chen H. Positive solutions of Neumann problems with singularities. J Math Anal Appl, 2008, 337(2):1267-1272
[10] Coelho I, Corsato C, Obersnel F, et al. Positive solutions of the Dirichlet problem for the one-dimensional Minkowski-curvature equation. Adv Nonlinear stud, 2012, 12(3):621-638
[11] Miciano A R, Shivaji R. Multiple positive solutions for a class of semipositone Neumann two point boundary value problems. J Math Anal Appl, 1993, 178(1):102-115
[12] Serra E, Tilli P. Monotonicity constraints and supercritical Neumann problems. Ann Inst H Poincaré Anal. Non Linéaire, 2011, 28(1):63-74
[13] Sun J P, Li W T. Multiple positive solutions to second-order Neumann boundary value problems. Appl Math Comput, 2003, 146(1):187-194
[14] Wei D Y. Exact number of solutions of a one-dimensional prescribed mean curvature equation with general nonlinearities. Acta Math Sci, 2016, 36A(1):1-13
Options
Outlines

/

〈 〉

Copyright © Editorial Office of Acta Mathematica Scientia
Add: P. O. Box 71010, Wuhan 430071, China　Tel: 027-87199206(中文版)　027-87199087(英文版)
E-mail: actams@wipm.ac.cn