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Advisor(s)
Abstract(s)
We start by studying the existence of positive solutions for the differential equation
u '' = a(x)u - g(u),
with u ''(0) = u(+infinity) = 0, where a is a positive function, and g is a power or a bounded function. In other words, we are concerned with even positive homoclinics of the differential equation. The main motivation is to check that some well-known results concerning the existence of homoclinics for the autonomous case (where a is constant) are also true for the non-autonomous equation.
This also motivates us to study the analogous fourth-order boundary value problem
{u((4)) - cu '' + a(x)u = vertical bar u vertical bar(p-1)u
u'(0) = u'''(0) = 0, u(+infinity) = u'(+infinity) = 0
for which we also find nontrivial (and, in some instances, positive) solutions.
Description
Keywords
Second order Fourth order Non-autonomous equation Variational methods Unbounded intervals Positive solution
Citation
Enguiça R, Gavioli A, Sanchez L. Solutions of second-order and fourth-order ODEs on the half-line.Nonlinear Analysis-Theory Methods & Applications. 2010; 73 (9): 2968-2979.