# Boundedness of solutions to a semilinear PDE

Let $$D$$ be the unit disk in $$\mathbb R^2$$ centered at the origin. Given any $$\lambda \in \mathbb R$$, let $$u_\lambda$$ be the unique solution to the semilinear elliptic equation $$-\Delta u + u^3=0 \quad \text{on D},$$ subject to the constant Dirichlet data $$u|_{\partial D} =\lambda$$. Prove that $$u(0)$$ is uniformly bounded in $$|\lambda|$$ and that more generally given any $$h\in (0,1)$$, the solution $$u(x)$$, with $$|x| is uniformly bounded in $$|\lambda|$$.

• I think this is true (maybe you already know it) since the radial ODE (with $u'(0)=0$, I guess) explodes for $u(0)$ large before reaching $r=1$. Do you have the same feeling? By the way, where does it come from? Dec 15, 2022 at 19:18
• That’s exactly my intuition for this problem. As for where it comes from I remember reading this in a paper but can not locate it anymore.
– Ali
Dec 15, 2022 at 19:54
• I just checked the details...maybe I should write an answer, it seems to work in any dimension Dec 15, 2022 at 19:58
• does maximum not show that the maximum is on the boundary? Dec 16, 2022 at 0:01

Fact 1. Let $$u'' \geq ku^\alpha$$ in $$[c,\ell[$$ with $$k>0, \alpha>1$$ and $$u,u' \geq 0$$. Let $$A=u(c)$$, then $$\ell \to c$$ as $$A \to \infty$$.
This follows by multiplying by $$u'$$ and integrating. One obtains $$u' \geq k \sqrt{u^{\alpha+1}-A^{\alpha+1}} \geq k \sqrt{(u-A)u^\alpha}$$ and integrating again $$\ell-c \leq k^{-1}\int_A^\infty \frac{ds}{\sqrt{(s-A)s^\alpha}} =k^{-1} \int_0^\infty \frac{dt}{(t+A)^{\frac \alpha 2}\sqrt t} \to 0$$ as $$A \to \infty$$.
Assume now that $$u''+\frac{N-1}{r}u'=u^3$$ with $$u'(0)=0$$ and $$u(1)=\lambda >0$$. The maximum principle yields $$u \geq 0$$ and $$r^{1-N} (r^{N-1} u')'=u^3$$ with $$u'(0)=0$$ also $$u' \geq 0$$. Multiplying the equation by $$u'$$ and integrating $$\frac 12 u'^2(r)+(N-1)\int_0^r \frac{u'^2}{t} dt= \frac 14 (u^4(r)-u(0)^4)$$ and then $$u'^2 \leq \frac 12 u^4$$. If $$A=u(0)$$ is sufficiently large, then $$u(r) \geq A$$ gives $$u'' \geq \frac 12 u^3$$ for $$r \geq \frac 12$$ and one concludes by Fact 1 with initial point $$\frac 12$$ that the solution explodes before reaching $$r=1$$.