On a singular boundary value problem arising in the theory of shallow membrane caps

Abstract We investigate the following singular boundary value problem which originates from the theory of shallow membrane caps, ( t 3 u ′ ( t ) ) ′ + t 3 ( 1 8 u 2 ( t ) − a 0 u ( t ) − b 0 t 2 γ − 4 ) = 0 , lim t → 0 + t 3 u ′ ( t ) = 0 , u ( 1 ) = 0 , where a 0 , b 0 , and γ are given constants. We show the existence of a positive solution to the above problem by means of a generalized lower and upper functions method involving limiting processes. We illustrate the theory by numerical experiments, in which we used the new version of the matlab code sbvp based on polynomial collocation, to approximate the solution of the membrane problem.

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