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 $Y\in H_0^1$?