ON THE EXISTENCE OF BUBBLE-TYPE SOLUTIONS OF NONLINEAR SINGULAR PROBLEMS

Feng Jiao; Jianshe Yu

doi:10.11948/2011015

2011 Volume 1 Issue 2

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Feng Jiao, Jianshe Yu. ON THE EXISTENCE OF BUBBLE-TYPE SOLUTIONS OF NONLINEAR SINGULAR PROBLEMS[J]. Journal of Applied Analysis & Computation, 2011, 1(2): 219-242. doi: 10.11948/2011015

Citation:

Feng Jiao, Jianshe Yu. ON THE EXISTENCE OF BUBBLE-TYPE SOLUTIONS OF NONLINEAR SINGULAR PROBLEMS[J]. Journal of Applied Analysis & Computation, 2011, 1(2): 219-242. doi: 10.11948/2011015

ON THE EXISTENCE OF BUBBLE-TYPE SOLUTIONS OF NONLINEAR SINGULAR PROBLEMS

College of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, People's Republic of China

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Abstract

Abstract

Considered in this paper is a class of singular boundary value problem, arising in hydrodynamics and nonlinear field theory, when centrally bubble-type solutions are sought:
(p(t)u')'=c(t)p(t)f(u), u'(0)=0, u(+∞)=L>0
in the half-line[0, +∞), where p(0)=0. We are interested in strictly increasing solutions of this problem in[0, ∞) having just one zero in (0, +∞) and finite limit at zero, which has great importance in applications or pure and applied mathematics. Sufficient conditions of the existence of such solutions are obtained by applying the critical point theory and by using shooting argument[1, 2] to better analysis the properties of certain solutions associated with the singular differential equation. To the authors' knowledge, for the first time, the above problem is dealt with when f satisfies non-Lipschitz condition. Recent results in the literature are generalized and significantly improved.
- Singular boundary value problem /
- Monotone solution /
- Variational methods /
- Unbounded domain /
- Homoclinic solution
MSC: 34B16;34B40;34C37;49J40

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References

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