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

doi:10.1016/S0252-9602(17)30085-1

数学物理学报(英文版) ›› 2017, Vol. 37 ›› Issue (5): 1472-1482.doi: 10.1016/S0252-9602(17)30085-1

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

马如云, 高红亮

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

收稿日期:2015-11-01 修回日期:2017-05-01 出版日期:2017-10-25 发布日期:2017-10-25
作者简介:Ruyun Ma,E-mail:mary@nwnu.edu.cn;Hongliang Gao,E-mail:gaohongliang101@163.com
基金资助:
Supported by the NSFC (11361054, 11671322)

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

Ruyun MA, Hongliang GAO

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Received:2015-11-01 Revised:2017-05-01 Online:2017-10-25 Published:2017-10-25
Supported by:
Supported by the NSFC (11361054, 11671322)

摘要/Abstract

摘要：

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.

关键词: multiple positive solutions, Neumann problem, prescribed mean curvature equation, time map

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

马如云, 高红亮. MULTIPLE POSITIVE SOLUTIONS FOR A CLASS OF SEMIPOSITONE NEUMANN PROBLEMS WITH SINGULAR φ-LAPLACIAN[J]. 数学物理学报(英文版), 2017, 37(5): 1472-1482.

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.

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

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

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