Existence of solutions for a quasilinear singular elliptic equation with Hardy potential and mixed boundary conditions

We mainly investigate the existence of solutions and parameter estimation for a class of quasilinear elliptic equations with singular Hardy potential and mixed boundary conditions. The differential operator in the equations originates from the study of self-trapped transverse magnetic TM-modes in cylindrical optical fibers made of self-focusing dielectric materials. In contrast to classical operators (e.g., the Laplacian, p-Laplacian, and Kirchhoff operators) that only depend on ∇u, the operator considered in this paper depends on both ∇u and u. By using variational methods, we establish the existence result of at least two weak solutions for the equation. Moreover, we derive the explicit range of parameter that guarantee the existence of such solutions.