Multiplicity of Positive Steady States for a Semipositone p-Laplacian Neumann Problem Arising in Radiolysis
DOI:
https://doi.org/10.22399/ijcesen.5306Keywords:
p-Laplacian operator, Semipositone problem, Neumann boundary Conditions, Quadrature method, Radiolysis, Reaction–diffusionAbstract
In this paper, we study the existence and multiplicity of positive solutions for the one-dimensional -Laplacian Neumann boundary value problem. The nonlinearity is semipositone since . Using the quadrature method (time-map method), we transform the problem into an algebraic equation involving a time-map function and establish conditions for the existence, uniqueness, and multiplicity of positive solutions depending on the parameter . In particular, we prove the existence of at least three positive solutions, and more generally, at least positive solutions for suitable ranges of . Numerical examples and direct simulations for both cases and are presented to illustrate the theoretical results. A physical interpretation is also provided within the framework of radiation chemistry: the problem describes the steady-state spatial distribution of reactive species generated by the radiolysis of confined aqueous systems under continuous irradiation, where the semipositone reaction term reflects the balance between radiation-induced production, recombination, and saturation, and the parameter encodes the dose rate. The multiple positive solutions correspond to distinct equilibrium concentration profiles attainable under identical irradiation conditions, a feature relevant to the accumulation of radiolytic products such as and in sealed irradiated systems. Our results extend existing works on semipositone Neumann problems and provide a clearer description of the bifurcation structure of positive solutions for the -Laplacian operator.
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