Relatively compact of minimizing sequences
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Let
$$
E(u)=frac12int_mathbb R^n |nabla u|^2-frac1p+1|u|^p+1 dx
$$
where $p$ is a positive constant. Define
$$
I_mu=inf.
$$
Assume $u_n$ is a minimizing sequence of $I_mu$, i.e.
$$
u_nin H^2(mathbb R^n),~~ ||u||_L^2^2=mu,~~
E(u_n)rightarrow I_mu.
$$
Why there are $y_nsubset mathbb R^n$ such that $u_n(cdot+y_n)$ is relatively compact
in $H^2(mathbb R^n)$ ?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
This question is a doubt when I read the 552th page of Cazenave, T.; Lions, Pierre-Louis, Orbital stability of standing waves for some nonlinear Schrödinger equations, Commun. Math. Phys. 85, 549-561 (1982). ZBL0513.35007.
functional-analysis pde banach-spaces
asked Sep 1 at 8:22
lanse7pty
1,7761723
add a comment |Â
up vote
2
down vote
favorite
Let
$$
E(u)=frac12int_mathbb R^n |nabla u|^2-frac1p+1|u|^p+1 dx
$$
where $p$ is a positive constant. Define
$$
I_mu=inf.
$$
Assume $u_n$ is a minimizing sequence of $I_mu$, i.e.
$$
u_nin H^2(mathbb R^n),~~ ||u||_L^2^2=mu,~~
E(u_n)rightarrow I_mu.
$$
Why there are $y_nsubset mathbb R^n$ such that $u_n(cdot+y_n)$ is relatively compact
in $H^2(mathbb R^n)$ ?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
This question is a doubt when I read the 552th page of Cazenave, T.; Lions, Pierre-Louis, Orbital stability of standing waves for some nonlinear Schrödinger equations, Commun. Math. Phys. 85, 549-561 (1982). ZBL0513.35007.
functional-analysis pde banach-spaces
asked Sep 1 at 8:22
lanse7pty
1,7761723
Maybe your question is related to the concentration-compactness principle.
– Rigel
Sep 1 at 8:37
@Rigel Could you talk about it detail ? I know the concentration-compactness principle. But how to use it ?
– lanse7pty
Sep 1 at 8:53
Unfortunately I'm not able to help you. My observation is related to the "sperimental" fact that, almost every time one gets a sequence of the form $u_n(cdot + y_n)$, then at some point the concentration-compactness principle comes into action.
– Rigel
Sep 1 at 8:58
@Rigel Thanks your hint too.
– lanse7pty
Sep 1 at 12:43
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Let
$$
E(u)=frac12int_mathbb R^n |nabla u|^2-frac1p+1|u|^p+1 dx
$$
where $p$ is a positive constant. Define
$$
I_mu=inf.
$$
Assume $u_n$ is a minimizing sequence of $I_mu$, i.e.
$$
u_nin H^2(mathbb R^n),~~ ||u||_L^2^2=mu,~~
E(u_n)rightarrow I_mu.
$$
Why there are $y_nsubset mathbb R^n$ such that $u_n(cdot+y_n)$ is relatively compact
in $H^2(mathbb R^n)$ ?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
This question is a doubt when I read the 552th page of Cazenave, T.; Lions, Pierre-Louis, Orbital stability of standing waves for some nonlinear Schrödinger equations, Commun. Math. Phys. 85, 549-561 (1982). ZBL0513.35007.
functional-analysis pde banach-spaces
asked Sep 1 at 8:22
lanse7pty
1,7761723
Let
$$
E(u)=frac12int_mathbb R^n |nabla u|^2-frac1p+1|u|^p+1 dx
$$
where $p$ is a positive constant. Define
$$
I_mu=inf.
$$
Assume $u_n$ is a minimizing sequence of $I_mu$, i.e.
$$
u_nin H^2(mathbb R^n),~~ ||u||_L^2^2=mu,~~
E(u_n)rightarrow I_mu.
$$
Why there are $y_nsubset mathbb R^n$ such that $u_n(cdot+y_n)$ is relatively compact
in $H^2(mathbb R^n)$ ?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
This question is a doubt when I read the 552th page of Cazenave, T.; Lions, Pierre-Louis, Orbital stability of standing waves for some nonlinear Schrödinger equations, Commun. Math. Phys. 85, 549-561 (1982). ZBL0513.35007.
functional-analysis pde banach-spaces
functional-analysis pde banach-spaces
asked Sep 1 at 8:22
lanse7pty
1,7761723
asked Sep 1 at 8:22
lanse7pty
1,7761723
asked Sep 1 at 8:22
lanse7pty
1,7761723
asked Sep 1 at 8:22
lanse7pty
1,7761723
asked Sep 1 at 8:22
lanse7pty
1,7761723
1,7761723
Maybe your question is related to the concentration-compactness principle.
– Rigel
Sep 1 at 8:37
@Rigel Could you talk about it detail ? I know the concentration-compactness principle. But how to use it ?
– lanse7pty
Sep 1 at 8:53
Unfortunately I'm not able to help you. My observation is related to the "sperimental" fact that, almost every time one gets a sequence of the form $u_n(cdot + y_n)$, then at some point the concentration-compactness principle comes into action.
– Rigel
Sep 1 at 8:58
@Rigel Thanks your hint too.
– lanse7pty
Sep 1 at 12:43
add a comment |Â
Maybe your question is related to the concentration-compactness principle.
– Rigel
Sep 1 at 8:37
@Rigel Could you talk about it detail ? I know the concentration-compactness principle. But how to use it ?
– lanse7pty
Sep 1 at 8:53
Unfortunately I'm not able to help you. My observation is related to the "sperimental" fact that, almost every time one gets a sequence of the form $u_n(cdot + y_n)$, then at some point the concentration-compactness principle comes into action.
– Rigel
Sep 1 at 8:58
@Rigel Thanks your hint too.
– lanse7pty
Sep 1 at 12:43
Maybe your question is related to the concentration-compactness principle.
– Rigel
Sep 1 at 8:37
Maybe your question is related to the concentration-compactness principle.
– Rigel
Sep 1 at 8:37
@Rigel Could you talk about it detail ? I know the concentration-compactness principle. But how to use it ?
– lanse7pty
Sep 1 at 8:53
@Rigel Could you talk about it detail ? I know the concentration-compactness principle. But how to use it ?
– lanse7pty
Sep 1 at 8:53
Unfortunately I'm not able to help you. My observation is related to the "sperimental" fact that, almost every time one gets a sequence of the form $u_n(cdot + y_n)$, then at some point the concentration-compactness principle comes into action.
– Rigel
Sep 1 at 8:58
Unfortunately I'm not able to help you. My observation is related to the "sperimental" fact that, almost every time one gets a sequence of the form $u_n(cdot + y_n)$, then at some point the concentration-compactness principle comes into action.
– Rigel
Sep 1 at 8:58
@Rigel Thanks your hint too.
– lanse7pty
Sep 1 at 12:43
@Rigel Thanks your hint too.
– lanse7pty
Sep 1 at 12:43
add a comment |Â
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Maybe your question is related to the concentration-compactness principle.
– Rigel
Sep 1 at 8:37
@Rigel Could you talk about it detail ? I know the concentration-compactness principle. But how to use it ?
– lanse7pty
Sep 1 at 8:53
Unfortunately I'm not able to help you. My observation is related to the "sperimental" fact that, almost every time one gets a sequence of the form $u_n(cdot + y_n)$, then at some point the concentration-compactness principle comes into action.
– Rigel
Sep 1 at 8:58
@Rigel Thanks your hint too.
– lanse7pty
Sep 1 at 12:43