Solvability of singular Dirichlet boundary-value problems with given maximal values for positive solutions

The singular boundary-value problem (g(x(t))) = muf (t, x(t), x(t)), x(0) = x(T) = 0 and max {x(t) : 0 less than or equal to t less than or equal to T} = A is considered. Here p is the parameter and the negative function f (t, u, v) satisfying local Caratheodory conditions on [0, T] x (0, infinity) x (R \ {0}) may be singular at the values u = 0 and v = 0 of the phase variables u and v. The paper presents conditions which guarantee that for any A 0 there exists muA 0 such that the above problem with mu = muA has a positive solution on (0, T). The proofs are based on the regularization and sequential techniques and use the Leray-Schauder degree and Vitalis convergence theorem.