POTW A Nonlinear Elliptic PDE on a Bounded Domain

[Euge]

Let $D$ be a smooth, bounded domain in $\mathbb{R}^n$ and $f : D \to (0, \infty)$ a continuous function. Prove that there exists no $C^2$-solution $u$ of the nonlinear elliptic problem $\Delta u^2 = f$ in $D$, $u = 0$ on $\partial D$.

 

[Infrared]

Spoiler

Start with Green's identity $\int_D \left(\psi \Delta \psi +||\nabla\psi||^2\right) dV=\int_{\partial D}\psi\nabla\psi\cdot dS.$

Substituting $\psi=u^2,$ we see that the lefthand side is strictly positive, but the righthand side would be zero if $u=0$ on the boundary.

 

[Euge]

Just adding a comment @Infrared's solution.

Spoiler

Since the two terms in the integrand on the left-hand side is nonnegative, if its integral over $D$ is zero, then both those terms are zero in $D$. In particular $fu^2 = 0$ in $D$, forcing $u = 0$ (since $f$ is positive). We get $f = \Delta u^2 = 0$, a contradiction. Therefore, the integral on the left-hand side is strictly positive.

 

[wrobel]

it is not clear enough what $u=0$ on $\partial D$ means. If $u\in C^2(D)\cap C(\overline D)$ then the assertion follows from the maximum principle directly; smoothness of $\partial D$ is not needed