A supercritical Sobolev type inequality in higher order Sobolev spaces and related higher order elliptic problems

A Sobolev type embedding for radially symmetric functions on the unit ball B in Rn, n≥3, into the variable exponent Lebesgue space L2⋆+|x|α(B), 2⋆=2n/(n−2), α>0, is known due to J.M. do Ó, B. Ruf, and P. Ubilla, namely, the inequalitysup⁡∫B|u(x)|2⋆+|x|αdx:u∈H0,rad1(B),‖∇u‖L2(B)=1<+∞ holds. In this work, we generalize the above inequality for higher order Sobolev spaces of radially symmetric functions on B, namely, the embeddingH0,radm(B)↪L2m⋆+|x|α(B) with 2≤m0 holds. Questions concerning the sharp constant for the inequality including the existence of the optimal functions are also studied. To illustrate the finding, an application to a boundary value problem on balls driven by polyharmonic operators is presented. This is the first in a set of our works concerning functional inequalities in the supercritical regime.