I have to solve PDE $\Delta u=u^3$ in $B(0,1)\subset\mathbb{R}^n$ with boundary condition $u=0$ on $\partial B(0,1)$. Is there an explicit expression for the solution? I know that the solution to Poisson equation $-\Delta u=f$ can be obtained by convolution with the fundamental solution as $$u(x)=\int_{\mathbb{R}^n}\Phi(x-y)f(y)\text{d}y .$$ But now I'm confused because the integral depends on $u$ itself. I tried to solve the radial version of this equation $$u^{\prime\prime}(r)+\frac{u^{\prime}(r)(n-1)}{r}-u^3(r)=0$$ but this seems impossible me to solve. And I'm not even sure if the equation is rotation invariant. Any help would be appreciated.


The only solution is $0$. Indeed, we multiply both sides of the equation $$ \Delta u=u^3 $$ with $-u$ and integrate by parts (Green's identity) to obtain that
$$ \int_{\Omega}|\nabla u|^2 dx= -\int u^4 \leq 0, $$ which implies that $\nabla u=0$ and hence $u\equiv C$ for some constant. It can only be $0$

  • $\begingroup$ Okay, now I got it, thanks a lot! $\endgroup$ – Infinitebig Sep 22 '17 at 15:02

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.