Existence of non-radially symmetric viscosity solutions to semilinear degenerate elliptic equations with radially symmetric coefficients in the plane, Part I

We prove the existence of non-radially symmetric solutions for semilinear degenerate elliptic equations with radially symmetric coefficients in the plane. We adapt the viscosity solution for the weak solution. The key arguments consist of the analysis of the structure of 2π-periodic solutions for the associated Laplace–Beltrami operator and construction of super- and sub-solutions which have the prescribed asymptotic structures.