Existence and Uniqueness of Positive Solutions for Singular Asymmetric Multi-Phase Equations

In this work, we establish the existence of positive solutions for a problem driven by a multi-phase operator composed of two distinct exponent Laplacian-type operators and a generalised lower-order term, which ensures asymmetric behaviour across three subregions of the domain under consideration. The reaction term involves a mild singularity at zero and includes a possibly sign-changing perturbation function. Under additional restrictive conditions, we also obtain a uniqueness result for the problem. Our existence result is based on pseudomonotone operator theory. Moreover, a detailed analysis, combined with a D & iacute;az-Sa & aacute;-type argument, allows us to also establish a uniqueness theorem. To the best of our knowledge, this is the first work addressing such a generalisation of the multi-phase operator. These novel results can serve as a foundation for more general physical and engineering models.