Existence of Multiple Solutions to a Transmission Problem

The transmission problem has emerged as an important component in various application fields, including industry, agriculture, and commerce. Simultaneously, it is also one of the most common challenges encountered in daily activities such as eating, drinking, living, and transportation. We have observed that numerous theories have been proposed regarding nonlocal transmission problems. However, how to find critical points of nonconvex functionals remains one of the major difficulties when one employs variational methods to establish the existence of solutions for nonlocal problems. This paper is devoted to the study of a class of weighted nonlocal transport equations. In order to establish the existence of solutions, we utilize algebraic analysis and combinatorial approximation techniques to overcome the nonconvexity of the corresponding Euler–Lagrange functional, and then prove the existence of a positive solution with the help of the strong maximum principle. Finally, by combining algebraic analysis with existing results, we obtain the existence of infinite solutions.