Phenomenon of bifurcation for nonlinear triharmonic equations

In this paper, the problem of bifurcation of the triharmonic equation is discussed by the best embedding constant and Nhari-type variation method. The harmonic equation has at least two solutions and its solution has bifurcation. The biharmonic equation has at least a radial positive solution, but bifurcation of the solution can´t be discovered. In triharmonic equation, the best embedding constant and Palais-Smale condition are given by the defined function, the triharmonic equation has at least a radial positive solution and a nonradial positive solution, bifurcation phenomenon can be discovered in the solution. We also supply method for discussing existence of solution of polyharmonic equation.