Qualitative properties of solutions to generalized eigenvalue problems

Purpose This paper investigates the qualitative properties of solutions to a fractional (p,)-Laplace eigenvalue problem with nonhomogeneous terms. We aim to establish non-existence results for specific parameter ranges prove the existence of weak solutions using variational methods and analyze their regularity. Additionally, we refine the Diaz–Saa inequality to derive a uniqueness result in the case r = q. Our study extends existing results in the field of fractional and nonlocal operators, providing new insights into their mathematical structure and potential applications. Design/methodology/approach We analyze a fractional (p,)-Laplace eigenvalue problem using variational methods and functional analysis techniques. Non-existence results are established via test function arguments. Existence of solutions is proved using the mountain pass theorem and minimization techniques. Regularity properties are derived using Hölder estimates, and a refined Diaz–Saa inequality is employed to establish uniqueness in the case r = q. The fractional Sobolev embedding theorem and compactness arguments play a crucial role in our analysis. This approach extends existing methods for nonlocal and nonhomogeneous operators. Findings This study establishes the existence, non-existence, uniqueness and regularity properties of weak solutions to a fractional (p,)-Laplace eigenvalue problem. We prove non-existence for certain values of λ and obtain existence results using variational methods and the mountain pass theorem. Furthermore, we refine the Diaz–Saa inequality to establish uniqueness when r = q. Regularity results are derived using Hölder continuity estimates. These findings provide new insights into the qualitative behavior of solutions and extend existing results for fractional and nonhomogeneous operators. Research limitations/implications One limitation of this study is that the analysis is restricted to a bounded domain Ω with a smooth boundary. Extending the results to more general domains, including those with irregular boundaries, remains an open problem. Moreover, the uniqueness result established here applies only to the case r = q, and further investigation is required to address the case r = q. From an applied perspective, the fractional (p,)-Laplacian model could also be explored in practical scenarios, such as optimization and image processing, where nonlocal effects play a significant role. Practical implications The results of this study contribute to the mathematical foundation of fractional and nonhomogeneous operators, which have applications in various fields, including physics, engineering and image processing. The fractional (p,)-Laplace model can be utilized in problems involving nonlocal diffusion, anomalous transport, and phase transitions. Moreover, the uniqueness and regularity results provide a rigorous basis for numerical approximations and computational methods in the applied sciences. Future research may explore extensions to irregular domains and real-world applications where nonlocal effects play a crucial role. Social implications The study of fractional (p,)-Laplace equations has potential societal impact in areas where nonlocal phenomena play a key role, such as medical imaging, material science, and environmental modeling. Improved understanding of these operators can enhance techniques in image reconstruction, tumor growth prediction, and porous media analysis. Furthermore, the mathematical framework developed in this work can contribute to advancements in computational methods used in engineering and technology, ultimately leading to more efficient solutions in real-world applications that affect daily life. Originality/value This study provides novel contributions to the analysis of fractional (p,)-Laplace eigenvalue problems by extending classical techniques to a nonhomogeneous and nonlocal setting. The refinement of the Diaz–Saa inequality and the uniqueness result for r = q represent significant advancements in the field. Moreover, the combination of variational methods, compactness arguments and fractional Sobolev embeddings offers new insights into the existence and regularity of weak solutions. These results not only enhance theoretical understanding but also provide a foundation for future research in applied mathematics and related disciplines.