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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.

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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 $0$

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

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