Let $\Omega \subset \mathbb{R}^N$, $N\geq 5$. Now assume the biharmonic problem with singular term as follow \begin{cases} \Delta^2u=\lambda \dfrac{u}{|x|^4}+u^{p} & \mathrm{in}\hspace{.2cm} \Omega \\ u=-\Delta u =0 & \mathrm{on}\hspace{.2cm} \partial \Omega \end{cases}

Where $\lambda=\dfrac{N^2(N-4)^2}{16}$ is the optimal constant of Rellich inequality and $p=\dfrac{N+4}{N-4}$ is the critical sobolev exponent. I know that there exist at least one positive solution but I am not able to show it by classical methods . (obviously $u \equiv 0$ is a trival solution).

I know that by using improved Hardy-Rellich inequality with remider term, we get

$$\|u\|^2= \int_{\Omega} \Big((\Delta u)^2 - \lambda \dfrac{u^2}{|x|^4}\Big) \,\mathrm{d}x $$

as an equivalent norm on the space $W_0^{2,2}(\Omega)$.

Now I define functional $J: W_0^{2,2}(\Omega) \to \mathbb{R} $ as follow

$$J(u)=\dfrac{1}{2} \int_{\Omega}(\Delta u)^2 - \dfrac{\lambda}{2} \int_{\Omega} \dfrac{u^2}{|x|^4} \,\mathrm{d}x - \dfrac{1}{p+1} \int_{\Omega} u^{p+1} \,\mathrm{d}x $$ I know critical points of this functional are weak solutions of problem.

By definition of the above norm I can write functional as follow.

$$J(u)=\dfrac{1}{2} \|u\|^2 - \dfrac{1}{p+1} \int_{\Omega} u^{p+1} \,\mathrm{d}x $$

If $1\leq p < \dfrac{N+4}{N-4} $ then By the help of semicontinuty of norm and compactness of imbedding $W_0^{2,2} (\Omega) \hookrightarrow L^{p}(\Omega)$ I can minimize this functional on the unite sphere of $L^{p+1}(\Omega)$ by classical methods of calculus of variation.

But $p=\dfrac{N+4}{N-4}$ is critical sobolev imbedding and the above argument break down.

Can some one help me to show the existence of at least one positive solution.

**So thanks for your help.**