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Let $\Omega\subseteq \mathbb{R}^2$ be a bounded domain with smooth boundary. Prove that, for all $p>1$ and $1\leq q<\infty$ for all $f\in L^{p}(\Omega)$. Then there exists a unique $u\in H_0^1(\Omega)$ such that $\Delta u=u^q+f$ in $\Omega.$

This is not a linear pde, so I'm looking for a standard proof of the existence of such a weak solution. About the uniqueness, I think energy estimate may work, but I also find it hard to establish such an estimate.

Any solution or hint is highly appreciated.

Hilton
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  • What is the boundary data? If $u$ vanishes then multiplying your PDE by $u$ and integrating by parts yields the energy functional $$E(u) = \int_{\Omega} \left( \frac{1}{2} \lvert \nabla u \rvert^{2} + \frac{u^{q+1}}{q+1} + fu \right) \ dx$$ and using this you can follow the same ideas as in here. – Matthew Cassell Apr 08 '21 at 17:28
  • The OP wants a weak solution in $H^1_0(\Omega)$ which in the strong setting corresponds to $u=0$ on $\partial \Omega$. – JackT Apr 09 '21 at 00:26
  • @Hilton I’m just wondering is there a reason why you think there exists a unique weak solution? It is possible that you are correct, but only because you are working in $\mathbb{R}^2$. If you were in $\mathbb{R}^n$, $n>2$ this is certainly not the case, which is why I ask. – JackT Apr 09 '21 at 00:40

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