Hi all  

I've the following non-linear PDE


$-\Delta Y + Y^3 =U$ on $\Omega \subset R^n $, open, bounded, Lipschitz boundary domain


$Y=0 , $ on $\partial\Omega$   



**1**.Let $Y\in H\_0^1 $ and as $H\_0^1 \hookrightarrow \hookrightarrow L^5 $ can we define a compact operator $T:L^5 \times [0,1] \rightarrow L^5 $  
and use the **Leray-Schauder Fixed Point Theorem** to prove the existence of a solution of above PDE for a general $U\in L^2$?


**2**.Or if not then how can we apply **Leray-Schauder Fixed Point Theorem** to proove existence of a solution in $Y\in H\_0^1$? 



Existence of solution of a Non-linear PDE via Fixed point theorem
The Leray-Schauder is a fixed point theorem in the spirit of Brouwer's. When it gives an existence result, it says nothing about uniqueness. Your problem is much better than that, because it does have a unique solution in $H^1\_0(\Omega)\cap L^4(\Omega)$, whenever $U$ belongs to the dual space $X=H^{-1}(\Omega)+L^{4/3}(\Omega)$. The reason is that $Y$ is a critical point of the functional
$$E[Y]=\int\_\Omega(\frac12|\nabla Y|^2+\frac14Y^4-UY)dx.$$
This function turns out to be continuous and coercive over $X$, and strictly convex. Therefore the standard arguments of the so-called *direct method of calculus of variations* yields existence and uniqueness of a critical point, which is the point $Y$ at which $E$ achieves its minimum.


