Nondegeneracy of the bubble solutions for critical equations involving the polyharmonic operator

Abstract

Abstract We reprove a result by Bartsch, Weth, and Willem (Calc. Var. Partial Differ. Equ. 18(3):253–268, 2003) concerning the nondegeneracy of bubble solutions for a critical semilinear elliptic equation involving the polyharmonic operator. The merit of our proof is that it does not rely on the comparison theorem. The argument of our proof mainly uses the stereographic projection with the Funk–Hecke formula, which works for general critical semilinear elliptic equations.

Keywords

• Nondegeneracy
• Polyharmonic operator
• Bubble solution
• Spherical harmonics