I am working on $p$-Laplace equation. that is $$\tag{1} -\text{div}(|\nabla u|^{p-2}\nabla u)=\delta_0 \;\; in\; \mathcal{D}'(\mathbb{R}^2), $$ and the $p$-bilaplace equation, that is $$\tag{2} \Delta(|\Delta u|^{p-2}\Delta u)=\delta_0 \;\; in\; \mathcal{D}'(\mathbb{R}^2), $$
In the case if $p=2$: The equation (1) can be written as $$ -\Delta u=\delta_0 \;\; in\; \mathcal{D}'(\mathbb{R}^2), $$ then its fundamental solution is $$\Gamma^1_2(x)=\frac1{2\pi}\ln(|x|),$$ More generally, in $\mathbb{R}^n$ the fundamental solution of equation (1) is
$$ \Gamma^1_p(x)=\begin{cases} \frac1{2\pi}\ln(|x|),&\; if\; p=n,\\ |x|,&\; if\; p=\infty,\\ |x|^{(p-n)/(p-1)},&\; otherwise. \end{cases} $$ (ref, book page 16).
If $p=2$, the second equation can be written as $$ \Delta^2 u=\delta_0 \;\; in\; \mathcal{D}'(\mathbb{R}^2), $$ then its fundamental solution is $$\Gamma^2_2(x)=-\frac{1}{8\pi}\left\lvert x\right\rvert^2\log\left\lvert x\right\rvert$$
I want to know the fundamental solution of tow equation (2) if $p\in ]1,2]$. I tried to prove it. I used the Green formula as following: $\forall \varphi \in \mathcal{D}(\mathbb{R}^2)$ $$\varphi(0)=\int_{\mathbb{R}^2}\Delta(|\Delta u|^{p-2}\Delta u)\varphi dx=\lim_{x\to\varepsilon}\int_{|x|>\varepsilon}|\Delta u|^{p-2}\Delta u\Delta\varphi dx-\lim_{x\to\varepsilon}\left(\int_{|x|=\varepsilon}|\Delta u|^{p-2}\Delta u\frac{\partial\varphi}{\partial n} d\sigma(x)-\int_{|x|=\varepsilon}|\Delta u|^{p-2}\frac{\partial(\Delta u)}{\partial n}\varphi d\sigma(x)\right)=...$$
please there is a hint to evalute the fundamental solution $\Gamma_p^2$ for the second equation (2).