Timeline for Uniqueness problem for an elliptic system
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
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| Apr 1, 2018 at 8:21 | comment | added | user539887 | It appears that, in the one-dimensional case (on $[0,A]$), the existence of a nontrivial solution is equivalent to the existence of a solution of $v'=u$, $v'=\frac{1}{d}(u-u^p)$ starting at the positive $u$-semiaxis and reaching the semiaxis for $x=A$. The system of ODEs is a Hamiltonian system, with $$H(u, v)=\frac{v^2}{2}-\frac{u^2}{2d} +\frac{u^{p+1}}{(p+1)d}.$$ And quite a lot is known on Hamiltonian systems (but I am no expert). To sum up, the problem boils down to that of (non)existence of periodic solutions of period $2A$. | |
| Apr 1, 2018 at 0:01 | comment | added | Gustave | But is there any conditions on p and q to expect uniqueness? | |
| Mar 31, 2018 at 16:11 | comment | added | Math604 | try googling Lin-Ni Conjecture... I think this might be related... | |
| Mar 31, 2018 at 7:44 | history | answered | user539887 | CC BY-SA 3.0 |