Interpolation inequality and some applications

We investigate {\bf explicit} universal estimate of finite Morse index solutions to polyharmonic equations. \,Differently to previous works \cite{BL2, DDF, fa, H1}, propose here a direct proof using a new interpolation inequality and a delicate boot-strap argument under large superlinear and subcritical growth conditions to show that the universal constant grows as a power function of the Morse index.\, Also, our interpolation inequality allows us to provide local $L^p$-$W^{2r,p}$ estimate.