Existence of solution of a Non-linear PDE via Fixed point theorem

Question

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 $Y\in H_0^1$?

Answer 1

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.

Answer 2

This equation has no unicity, and it has at least a second solution by using the Mountain Pass Theorem, check Evans, PDE, pp 482-486.