I am going through the proof of Gagliardo-Nirenberg interpolation inequality, in A First Course in Sobolev Spaces, by Giovanni Leoni, Chapter 12. Here I am stuck with a Exercise problem comes in the middle of the proof (page 376):
Let $u\in \dot{W}^{1,N}(\mathbb{R}^N)$, $N\geq 2$, with $u\in L^q(\mathbb{R}^N)$ for some $1\leq q<\infty$. Let $v:=|u|^t$, where $t>1$ is such that $v\in W^{1,1}(\mathbb{R}^N)$. Prove that $$\|u\|_{L^{tN/(N-1)}}\leq c\|u\|_{L^{(t-1)N/(N-1)}}^{(t-1)/t}\|\nabla u\|_{L^N}^{1/t}.$$
Here $\dot{W}^{1,N}$ denots homogeneous Sobolev space.
My best try was that, by using $L^p$ type interpolation inequality to $\|v\|_{L^{N/(N-1)}}$ as $\frac{1}{\frac{N}{N-1}}=\frac{1-\frac{1}{t}}{\left(1-\frac{1}{t}\right)\frac{N}{N-1}}+\frac{\frac{1}{t}}{\infty}$, $$\|u\|_{L^{tN/(N-1)}}=\|v\|_{L^{N/(N-1)}}^{1/t}\leq\|v\|_{L^{(1-1/t)N/(N-1)}}^{(1-1/t)1/t}\| v\|_{L^\infty}^{1/t^2}\\=\|u\|_{L^{(t-1)N/(N-1)}}^{(t-1)/t}\| v\|_{L^\infty}^{1/t^2}.$$ I believe such $v\in L^\infty$, however I cannot proceed further than this. I will deeply appreciate if somebody can provide some help.
Assume $u\in C_0^\infty({\mathbb R}^N)$, apply the classical Sobolev inequality $$ \Big(\int_{{\mathbb R}^N} (u^t)^{\frac{N}{N-1}}\Big)^{\frac{N-1}{N}} \leq C\int_{{\mathbb R}^N} |\nabla(u^t)|, $$ then apply Holder on the right hand side: $$ \int_{{\mathbb R}^N} |\nabla(u^t)|=t\int_{{\mathbb R}^N} |u^{t-1}\nabla u| \leq t\Big(\int_{{\mathbb R}^N} |u^{t-1}|^{\frac{N}{N-1}}\Big)^{\frac{N-1}{N}}\Big( \int_{{\mathbb R}^N} |\nabla u|^N\Big)^{\frac{1}{N}}, $$ thus $$ \|u\|_{\frac{tN}{N-1}}^t\leq Ct\|u\|_{\frac{(t-1)N}{N-1}}^{t-1}\cdot \|\nabla u\|_N. $$ Take $\frac 1t$ power.