Let $f \in L^2(\mathbb{R}^n).$ Why the equation $\Delta u - u = \dfrac{\partial f}{\partial x_i}$ admits a unique solution $u \in H^1(\mathbb{R}^n)$?
2- Prove that there exist a constant $C \geq 0$ that $||u||_{H^1} \leq C ||f||_{L^2}$.
3- Prove that there exist a constant $M \geq 0$ that for all $u \in H^2(\mathbb{R}^n)$ we have $||u||_{H^2} \leq M (||u||_{L^2})$.
4- We assume that $$\sum_{i,j=1=1,\dots,n} \displaystyle\int_{\mathbb{R}^n} \dfrac{\partial^2 u}{\partial x_i^2} \overline{\dfrac{\partial^2 v}{\partial x_j^2}} dx + \lambda \displaystyle\int_{\mathbb{R}^n} u \overline{v} dx$$ represente an scalar product to $H^2(\mathbb{R}^n)$ for all $\lambda > 0.$
Prove that this scalar product is equivalent to the classical scalar product to $H^2(\mathbb{R}^n)$
My problem is the question 4. How we can prove the equivalence of this two scalar product? Thank's.
One can prove (using the Fourier transform, e.g.) that $$ \lVert u\rVert_{H^2} \le C(\lVert u \rVert_{L^2} + \lVert \Delta u\rVert_{L^2}).$$ (This might be what you meant in item 3 - what's currently written is trivially true.)
Then for (4) you want to show that the usual norm on $H^2$ is equivalent to the norm $$ \lVert u\rVert_{\tilde{H}^2}^2 := \int_{\mathbb{R}^n} \lvert \Delta u\rvert^2 + \lvert u \rvert^2$$ i.e. that there exist constants $C_1,C_2>0$ so that $$ C_1 \lVert u\rVert_{\tilde{H}^2}^2 \le \lVert u\rVert_{H^2}^2 \le C_2 \lVert u\rVert_{\tilde{H}^2}^2.$$ The second inequality follows from above (from 3, probably), and the first follows easily from the estimate $$ \int_{\mathbb{R}^n} \lvert \Delta u\rvert^2 \le C \int_{\mathbb{R}^n} \sum_{ij}\lvert \partial_i \partial_j u\rvert^2$$