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

## TL;DR

This paper extends Sobolev embedding inequalities for radially symmetric functions to higher order Sobolev spaces, investigates sharp constants and optimal functions, and applies results to polyharmonic boundary value problems.

## Contribution

It generalizes known supercritical Sobolev inequalities to higher order spaces and explores their sharpness and applications.

## Key findings

- Established higher order Sobolev embeddings with supercritical exponents.
- Analyzed the existence of sharp constants and optimal functions.
- Applied the inequalities to polyharmonic boundary value problems.

## Abstract

A Sobolev type embedding for radially symmetric functions on the unit ball $B$ in $\mathbb R^n$, $n\geq 3$, 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 \'O, B. Ruf, and P. Ubilla, namely, the inequality \[ \sup\Big\{\int_B |u(x)|^{2^\star+|x|^\alpha} dx : u\in H^1_{0,{\rm rad}}(B), \|\nabla u\|_{L^2(B)} =1\Big\} < +\infty \] holds. In this work, we generalize the above inequality for higher order Sobolev spaces of radially symmetric functions on $B$, namely, the embedding \[ H^m_{0,{\rm rad}}(B) \hookrightarrow L_{2_m^\star + |x|^\alpha} (B) \] with $2\leq m < n/2$, $2_m^* = 2n/(n-2m)$, 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.