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

Abstract

A Sobolev type embedding for radially symmetric functions on the unit ball $B$ in ${\mathbb{R}}^n$, $n\ge 3$, into the variable exponent Lebesgue space $L_{2^{\star }+|x{|}^{\alpha }}(B)$, $2^{\star }=2n/(n-2)$, $\alpha >0$, is known due to J.M. do Ó, B. Ruf, and P. Ubilla, namely, the inequality \[ \sup\Big\{\int_B |u(x)|^{2^\star+|x|^α} dx : u\in H^1_{0,{\rm rad}}(B), \|\nabla u\|_{L^2(B)} =1\Big\} < +\infty \] holds. In this work, we generalize the above inequality for higher order Sobolev spaces of radially symmetric functions on $B$, namely, the embedding \[ H^m_{0,{\rm rad}}(B) \hookrightarrow L_{2_m^\star + |x|^α} (B) \] with $2\le m<n/2$, $2_m^{\ast }=2n/(n-2m)$, and $\alpha >0$ 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.

26 pages, 0 figure