Let $N \geq 2$ an natural number and let $k>0$. Consider de equation defined on $\mathbb{R}^N\setminus \{0\}:$ $$\Delta u(x)-k^2u(x)=0.$$
It is known that if $k=1$ then $E_N(x)=\gamma_N |x|^{1-N/2}K_{\frac{N}{2}-1}(|x|)$ is solution of the equation above where $\gamma_N =(2 \pi)^{-N/2}$ and $K_{\frac{N}{2}-1}$ is the modified Bessel function of second kind of order $N/2-1$. for general $k>0$ is an fact that $E_N^k(x)=k^{N-2}E_N(kx)$ solves the above equation too. In what follows $l_q:=2^{q-1}\Gamma(q)$. Here are some facts about the fundamental solution $E_N^k$:
- $E_N^k \in C^{\infty}(\mathbb{R}^N \setminus \{0\}),$
- $E_N^k(x)>0,$
- $\lim_{|x| \to 0}\frac{E_2^k(x)}{-\ln(k|x|)}=\frac{1}{2 \pi l_1},$
- For $N \geq3$ follows that $\lim_{|x|\to 0}|x|^{N-2}E_N^{k}(x)=\gamma_N l_{\frac{N}{2}-1},$
- For $N \geq2$ follows that $\lim_{|x| \to 0}|x|^{N-1}|\nabla E_N^k(x)|=\gamma_N l_{\frac{N}{2}},$
- $\lim_{|x| \to \infty}e^{k|x|}E_N^{k}(x)=0$ and $\lim_{|x| \to \infty}e^{k |x|}|\nabla E_N^{k}(x)|=0.$
The properties above they are all very clear for me and with a lot of work I was able to prove each one of them. What I wish to understand is the following: How can I use the properties of $E_N^k$ stated above to prove that
- $E_2^k \in L^p(\mathbb{R}^N)$ for $1 \leq p <\infty,$
- $E_N^k \in L^{p}(\mathbb{R}^N)$ for $1 \leq p <\frac{N}{N-2}$ for $N \geq3,$
- $\partial_iE_N^{k} \in L^{p}(\mathbb{R}^N)$ for $1 \leq p<\frac{N}{N-1}$ for $N \geq 2$ and $i=1,2,\dots,N.$
I think that is useful in this context the use of the coarea formula which can be found here:https://en.wikipedia.org/wiki/Coarea_formula. But i'm having no success in the application of the formula because I'm having difficulty in using the properties above, the reason is I have no idea on how to make the terms $|x|^{N-2}$ or $e^{k |x|}$ to appear and apply the limits above. That is the reason why I need help and hints on how to proceed, every idea will be very useful and I will read and learn with care, thank you so much for the attention.