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Let $\Omega$ be a bounded domain in $\mathbb R^3$ with a smooth boundary. Consider a smooth real valued function $F:\overline\Omega \times \mathbb R \to \mathbb R$ with the property that $\partial_s F(x,s)\geq 0$ for all $x \in \Omega$ and all $s\in \mathbb R$. Consider the following boundary value problem $$-\Delta u + F(x,u) =0 \quad \text{on $\Omega$} \quad \text{subject to $u|_{\partial \Omega}=f$}.$$

It is well known that the above problem is well posed in the sense that given any $f \in C^{2,\alpha}(\partial \Omega)$ the above problem admits a unique solution in $C^{2,\alpha}(\Omega)$.

My question is as follows: Given a general nonlinear function $F$ as above and any $p\in \Omega$ and any $\lambda>0$ is it true that there exists $f \in C^{2,\alpha}(\partial \Omega)$ such that the solution $u$ to the above elliptic equation satisfies: $|u(p)|>\lambda$? Even the case of nonlinear functions $F(x,s)= a(x) s^{2k+1}$ with $a$ positive and $k\geq 1$ would be very interesting for me.

Thanks,

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  • $\begingroup$ If we put your equation in the standard form (i.e. as a positive definite form) $$ G(\partial_{i,j} u, \partial_i u, u, x) = \Delta u - F(x,u) = 0 $$ then the result follows from a standard maximum principle, but perhaps I am missing something. $\endgroup$
    – Daniele Tampieri
    Commented Aug 6, 2022 at 20:16
  • $\begingroup$ I want the solution at any one point inside to be arbitrarily large for some dirichlet data. How does this follow from maximum principle? $\endgroup$
    – Ali
    Commented Aug 7, 2022 at 9:50
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    $\begingroup$ It is not about attaining maximum inside. For instance think about this example: you consider the p to be the origin and then the question asks whether you can cook up a solution that is of value bigger than \lambda at the origin for lambda very large. It is not beyond unreasonable that if you put Dirichlet data 100\lambda on the boundary then maybe the solution at the origin will be of size roughly lambda. This would not contradict maximum principle. $\endgroup$
    – Ali
    Commented Aug 8, 2022 at 16:39
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    $\begingroup$ To summarize I think your confusion comes because you think I am asking for just one f that does the job. But f depends on p and lambda. $\endgroup$
    – Ali
    Commented Aug 8, 2022 at 16:42
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    $\begingroup$ I have read again your question and I saw my mistake. You need a Dirichlet data $f$ such that, for an arbitrarily chosen $p\in\Omega\setminus\partial\Omega$, $|u(p)|>\lambda$. These results are commonly known as pointwise estimates: I googled it a bit and I found only a work by Kresin and Maz'ya where the gradient of the solution of a parabolic equation is estimated by the value of the normal derivative at the boundary, and even in this case it is a bound tout court, not a lower bound. Perhaps, asking to Vladimir Maz'ya would be a nice idea. $\endgroup$
    – Daniele Tampieri
    Commented Aug 13, 2022 at 19:09

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