Assume that is a positive continuous function in and satisfies some suitable conditions. We prove that the quasilinear elliptic equation in admits at least two solutions in (one is a positive ground-state solution and the other is a sign-changing solution).
Huei-li Lin. "Multiple Solutions of Quasilinear Elliptic Equations in ." Int. J. Differ. Equ. 2010 (SI3) 1 - 12, 2010. https://doi.org/10.1155/2010/673526