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This work is devoted to the existence of positive, vanishing nontrivial solutions for singular boundary value problems on the half-line. The nonlinearity, which encompasses some physical laws, depends on the solution and its derivative, may exhibit a singularity at the origin in its second argument and is further allowed to change sign. Existence results are proved using two different methods: the upper and lower solutions technique and the fixed point index on cones of weighted Banach spaces. The singularity is treated by means of regularization, approximation and compactness arguments. The problem originates from chemistry and biology.

In this paper, we investigate the existence of positive solution for a class of singular third-order boundary value problem associated with a φ-Laplacian operator and posed on the positive half-line:where µ ≥ 0. By using the upper and lower solution approach and the fixed point theory, the existence of positive solutions is proved under a monotonic condition on f. The nonlinearity f may be singular at x = 0. An example of application is included to illustrate the main existence result. RESUMENEn este artículo investigamos la existencia de una solución positiva de una clase de problema singular de valores de frontera de tercer-orden asociado con el operador φ-Laplaciano y colocado sobre la semirecta real positiva:donde µ ≥ 0. Usando la técnica de sub y súper soluciones y la teoría del punto fijo, se prueba la existencia de soluciones positivas bajo una condición de monotonicidad sobre f. La nolinealidad f puede ser singular en x = 0. Se incluye un ejemplo de aplicación para ilustrar el resultado principal de existencia.

This work is devoted to the existence of solutions for a class of singular third-order boundary value problem associated with a φ-Laplacian operator and posed on the positive half-line; the nonlinearity also depends on the first derivative. The upper and lower solution method combined with the fixed point theory guarantee the existence of positive solutions when the nonlinearity is monotonic with respect to its arguments and may have a space singularity; however no Nagumo type condition is assumed. An example of application illustrates the applicability of the existence result.

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