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 ${\mathbb{R}}^n$, $n\ge 3$, into the variable exponent Lebesgue space $L_{2^{\star }+\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }}(B)$, $2^{\star }=2n\mathrm{/}(n-2)$, $\alpha >0$, is known due to J.M. do 'O, B. Ruf, and P. Ubilla, namely, the inequality $$\sup \{{\int }_B\mathrm{\mid }u(x){\mathrm{\mid }}^{2^{\star }+\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }}dx:u\in H_{0,\mathrm{r}\mathrm{a}\mathrm{d}}^1(B),\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{L^2(B)}=1\}<+\mathrm{\infty }$$ holds. In this work, we generalize the above inequality for higher order Sobolev spaces of radially symmetric functions on $B$, namely, the embedding $$H_{0,\mathrm{r}\mathrm{a}\mathrm{d}}^m(B)\hookrightarrow L_{2_m^{\star }+\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }}(B)$$ with $2\le m<n\mathrm{/}2$, $2_m^{\ast }=2n\mathrm{/}(n-2m)$, 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.