Strong singularities in mixed boundary
value problems

Abstract: We study singular boundary value problems with mixed boundary conditions of the form
(p(t)u')'+ p(t)f(t,u,p(t)u')=0, \quad \lim _{t\to 0+}p(t)u'(t)=0, \quad u(T)=0,
where $[0,T]\subset \R .$ We assume that $\D \subset \R ^2,$ $f$ satisfies the Carathéodory conditions on $(0,T)\times \D ,$ $p\in C[0,T]$ and ${1/p}$ need not be integrable on $[0,T].$ Here $f$ can have time singularities at $t=0$ and/or $t=T$ and a space singularity at $x=0$. Moreover, $f$ can change its sign. Provided $f$ is nonnegative it can have even a space singularity at $y=0.$ We present conditions for the existence of solutions positive on $[0,T).$

Keywords: singular mixed boundary value problem, positive solution, lower function, upper function, convergence of approximate regular problems

Classification (MSC2000): 34B16, 34B18

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