Mixing
ODE arising from best constant for Sobolev Embedding [closed]
Clash Royale CLAN TAG#URR8PPP
up vote
0
down vote
favorite
I was trying to solve the critical exponent PDE
$$Delta u = u^fracn+2n-2$$
in $mathbbR^n$ and ultimately reduced it to the following ODE $$fracdydr=ce^int_0^ry-y^2-fracyr$$
with the initial condition $y(0)=k in mathbbR$ for any $ngeq 3$. I want to prove existence and uniqueness in $[0,epsilon)$. I tried to do this for the simpler case
$$fracdydr=1-y^2-fracyr$$
but even here I don't know how to deal with the $1/r$. Any hints?
differential-equations pde
closed as off-topic by Alex Francisco, Taroccoesbrocco, Brian Borchers, José Carlos Santos, Parcly Taxel Jul 17 at 0:54
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." РAlex Francisco, Taroccoesbrocco, Brian Borchers, Jos̩ Carlos Santos, Parcly Taxel
add a comment |Â
up vote
0
down vote
favorite
I was trying to solve the critical exponent PDE
$$Delta u = u^fracn+2n-2$$
in $mathbbR^n$ and ultimately reduced it to the following ODE $$fracdydr=ce^int_0^ry-y^2-fracyr$$
with the initial condition $y(0)=k in mathbbR$ for any $ngeq 3$. I want to prove existence and uniqueness in $[0,epsilon)$. I tried to do this for the simpler case
$$fracdydr=1-y^2-fracyr$$
but even here I don't know how to deal with the $1/r$. Any hints?
differential-equations pde
closed as off-topic by Alex Francisco, Taroccoesbrocco, Brian Borchers, José Carlos Santos, Parcly Taxel Jul 17 at 0:54
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." РAlex Francisco, Taroccoesbrocco, Brian Borchers, Jos̩ Carlos Santos, Parcly Taxel
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I was trying to solve the critical exponent PDE
$$Delta u = u^fracn+2n-2$$
in $mathbbR^n$ and ultimately reduced it to the following ODE $$fracdydr=ce^int_0^ry-y^2-fracyr$$
with the initial condition $y(0)=k in mathbbR$ for any $ngeq 3$. I want to prove existence and uniqueness in $[0,epsilon)$. I tried to do this for the simpler case
$$fracdydr=1-y^2-fracyr$$
but even here I don't know how to deal with the $1/r$. Any hints?
differential-equations pde
I was trying to solve the critical exponent PDE
$$Delta u = u^fracn+2n-2$$
in $mathbbR^n$ and ultimately reduced it to the following ODE $$fracdydr=ce^int_0^ry-y^2-fracyr$$
with the initial condition $y(0)=k in mathbbR$ for any $ngeq 3$. I want to prove existence and uniqueness in $[0,epsilon)$. I tried to do this for the simpler case
$$fracdydr=1-y^2-fracyr$$
but even here I don't know how to deal with the $1/r$. Any hints?
differential-equations pde
edited Jul 16 at 12:02
asked Jul 16 at 10:59
user456218
closed as off-topic by Alex Francisco, Taroccoesbrocco, Brian Borchers, José Carlos Santos, Parcly Taxel Jul 17 at 0:54
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." РAlex Francisco, Taroccoesbrocco, Brian Borchers, Jos̩ Carlos Santos, Parcly Taxel
closed as off-topic by Alex Francisco, Taroccoesbrocco, Brian Borchers, José Carlos Santos, Parcly Taxel Jul 17 at 0:54
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." РAlex Francisco, Taroccoesbrocco, Brian Borchers, Jos̩ Carlos Santos, Parcly Taxel
add a comment |Â
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
1
down vote
Hint.
Making the variable change
$$
y = fracz'zRightarrow r z''+z'-r z = 0
$$
which is linear. All this for $r z ne 0$
answered Jul 16 at 11:10
Cesareo
5,7922412
This is the ODE I started with. Can you solve it? Because the 1/r is there as well.
– user456218
Jul 16 at 11:40
1
@AdityaKumar The solution is $y(r) = C_1 J_0(i r)+C_2 Y_0(-i r)$ (Bessel functions)
– Cesareo
Jul 16 at 11:44
Okay, but this won't work in the general case :(
– user456218
Jul 16 at 11:54
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Hint.
Making the variable change
$$
y = fracz'zRightarrow r z''+z'-r z = 0
$$
which is linear. All this for $r z ne 0$
answered Jul 16 at 11:10
Cesareo
5,7922412
This is the ODE I started with. Can you solve it? Because the 1/r is there as well.
– user456218
Jul 16 at 11:40
1
@AdityaKumar The solution is $y(r) = C_1 J_0(i r)+C_2 Y_0(-i r)$ (Bessel functions)
– Cesareo
Jul 16 at 11:44
Okay, but this won't work in the general case :(
– user456218
Jul 16 at 11:54
add a comment |Â
up vote
1
down vote
Hint.
Making the variable change
$$
y = fracz'zRightarrow r z''+z'-r z = 0
$$
which is linear. All this for $r z ne 0$
answered Jul 16 at 11:10
Cesareo
5,7922412
This is the ODE I started with. Can you solve it? Because the 1/r is there as well.
– user456218
Jul 16 at 11:40
1
@AdityaKumar The solution is $y(r) = C_1 J_0(i r)+C_2 Y_0(-i r)$ (Bessel functions)
– Cesareo
Jul 16 at 11:44
Okay, but this won't work in the general case :(
– user456218
Jul 16 at 11:54
add a comment |Â
up vote
1
down vote
up vote
1
down vote
Hint.
Making the variable change
$$
y = fracz'zRightarrow r z''+z'-r z = 0
$$
which is linear. All this for $r z ne 0$
answered Jul 16 at 11:10
Cesareo
5,7922412
Hint.
Making the variable change
$$
y = fracz'zRightarrow r z''+z'-r z = 0
$$
which is linear. All this for $r z ne 0$
answered Jul 16 at 11:10
Cesareo
5,7922412
answered Jul 16 at 11:10
Cesareo
5,7922412
answered Jul 16 at 11:10
Cesareo
5,7922412
answered Jul 16 at 11:10
Cesareo
5,7922412
5,7922412
This is the ODE I started with. Can you solve it? Because the 1/r is there as well.
– user456218
Jul 16 at 11:40
1
@AdityaKumar The solution is $y(r) = C_1 J_0(i r)+C_2 Y_0(-i r)$ (Bessel functions)
– Cesareo
Jul 16 at 11:44
Okay, but this won't work in the general case :(
– user456218
Jul 16 at 11:54
add a comment |Â
This is the ODE I started with. Can you solve it? Because the 1/r is there as well.
– user456218
Jul 16 at 11:40
1
@AdityaKumar The solution is $y(r) = C_1 J_0(i r)+C_2 Y_0(-i r)$ (Bessel functions)
– Cesareo
Jul 16 at 11:44
Okay, but this won't work in the general case :(
– user456218
Jul 16 at 11:54
This is the ODE I started with. Can you solve it? Because the 1/r is there as well.
– user456218
Jul 16 at 11:40
This is the ODE I started with. Can you solve it? Because the 1/r is there as well.
– user456218
Jul 16 at 11:40
1
1
@AdityaKumar The solution is $y(r) = C_1 J_0(i r)+C_2 Y_0(-i r)$ (Bessel functions)
– Cesareo
Jul 16 at 11:44
@AdityaKumar The solution is $y(r) = C_1 J_0(i r)+C_2 Y_0(-i r)$ (Bessel functions)
– Cesareo
Jul 16 at 11:44
Okay, but this won't work in the general case :(
– user456218
Jul 16 at 11:54
Okay, but this won't work in the general case :(
– user456218
Jul 16 at 11:54
add a comment |Â