Timeline for Non-existence of rapidly decaying solutions of certain elliptic semilinear equations
Current License: CC BY-SA 4.0
11 events
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| Mar 1, 2022 at 22:08 | comment | added | Willie Wong | For suitably strong weights, and suitably large $m$, maybe? You have to compute. (Essentially you need to prove that the associated bilinear form is coercive; and the enemy is something that looks like $\int \partial w u \cdot \partial u$ where $w$ is the weight. You can certainly cook up function spaces in which, for large enough $m$, every thing works out nicely.) | |
| Mar 1, 2022 at 21:54 | comment | added | S.Z. | @WillieWong, I see. But then could the argument in your answer work for functions of polynomial growth if one introduces some weights? In other words, is there a chance for the operator $-\Delta+1$ to be (strictly) positive on a weighted $L^2$-space? | |
| Mar 1, 2022 at 21:30 | comment | added | Willie Wong | Well, be careful. Agmon estimates usually assume solution has some decay at infinity to start with (something like $L^p$ for some $p$). (It says something along the line that if a solution is in $L^p$ then it decays exponentially.) Solutions of polynomial growth is a different issue. | |
| Mar 1, 2022 at 20:53 | vote | accept | S.Z. | ||
| Mar 1, 2022 at 20:52 | comment | added | S.Z. | @WillieWong, thanks a lot. Actually I'm interested in non-existence of classical solutions of polynomial growth at infinity. I asked the question for rapidly decaying functions because I thought that's simpler. So these Agmon estimates, together with your answer, seems to be exactly what I need. | |
| Mar 1, 2022 at 19:35 | comment | added | Willie Wong | If you are interested: the standard reference for Agmon estimates is Agmon, S., Lectures on Exponential Decay of Solution of Second-Order Elliptic Equation, Princeton University Press, 1982. | |
| Mar 1, 2022 at 19:33 | comment | added | Willie Wong | Incidentally, the question as you posed is a bit strange: by virtue of Agmon estimates, frequently when you have a solution to the equation of the type you stated, the solution would have exponential (and hence rapid) decay. So if you are primarily interested in decay rates, to rule out rapidly decaying solutions is more or less the same as looking for conditions to make the equation not have any non-trivial solutions. But your question doesn't seem to be focused on regularity issues. | |
| Mar 1, 2022 at 19:20 | answer | added | Willie Wong | timeline score: 5 | |
| Mar 1, 2022 at 19:06 | comment | added | S.Z. | @WillieWong, actually I'm interested in this case. Can you provide a reference for it? | |
| Mar 1, 2022 at 17:58 | comment | added | Willie Wong | Well... there's the case when $p$ is odd and $\lambda \geq 0$.... (but I guess that's too trivial for you) | |
| Mar 1, 2022 at 17:33 | history | asked | S.Z. | CC BY-SA 4.0 |