Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 105 / 1 / c / i / Solution

For $1\le p<n$, define the Sobolev conjugate exponentThe Sobolev inequality, also called the Gagliardo--Nirenberg--Sobolev inequality, states that there is a constant $C=C(n,p)$ such thatfor every $u\in C_c^{\infty }({\mathbb{R}}^n)$, and hence by completion for every $u\in W^{1,p}({\mathbb{R}}^n)$ for which the right formulation applies.