Reference about eigenvalue of Kirchhoff type operator
Assume $u_0$ is a positive radial symmetric nontrival solution of
begin{align}
-A(||nabla u||^2)Delta u + lambda u - |u|^2u=0 tag{1} \
uin H^2( R^n)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
end{align}
where $A(s)$ is a given positve smooth function, $lambda >0$ is a constant. The real part of linear operator of (1) at $u_0$ is
$$
Lu=-2A'(||nabla u_0||^2) Delta u_0 int nabla u_0 cdot nabla u dx
-A(||nabla u_0||^2)Delta u +lambda u-3u_0^2 u
$$
In fact, I want to know the eigenvalue of $L$, but seemly, there is not any similar work. So, firstly, I want to know the eigenvalue of
$$
Hu=-A(||nabla u_0||^2)Delta u +lambda u-3u_0^2 u.
$$
In fact, I found a similar work. In Stability of semiclassical bound states of nonlinear Schrödinger equations with potentials, the author give a lemma as picture below
The operator $L_+^0$ is similar to $H$, but I still can't get the negative eigenvalue and kernel of $H$, and its eigenfunction respectly, since the $u_0$ in Lemma 3.1 is a given function ($u_0(x)=sqrt{2lambda}sech sqrt{2lambda }x$). In my question , the $u_0$ is not so good. I want to deal this question, what paper or book I should read ? Thanks for any help.
PS: Reference of Lemma 3.1 is
Titchmarsh, E. C., Eigenfunction expansions associated with second-order differential equations, Oxford: At the Clarendon Press. 175 p. (1946). ZBL0061.13505.
but I don't found the lemma 3.1 in it.
pde elliptic-equations
asked Nov 18 at 14:12
lanse7pty
1,7751823
add a comment |
Assume $u_0$ is a positive radial symmetric nontrival solution of
begin{align}
-A(||nabla u||^2)Delta u + lambda u - |u|^2u=0 tag{1} \
uin H^2( R^n)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
end{align}
where $A(s)$ is a given positve smooth function, $lambda >0$ is a constant. The real part of linear operator of (1) at $u_0$ is
$$
Lu=-2A'(||nabla u_0||^2) Delta u_0 int nabla u_0 cdot nabla u dx
-A(||nabla u_0||^2)Delta u +lambda u-3u_0^2 u
$$
In fact, I want to know the eigenvalue of $L$, but seemly, there is not any similar work. So, firstly, I want to know the eigenvalue of
$$
Hu=-A(||nabla u_0||^2)Delta u +lambda u-3u_0^2 u.
$$
In fact, I found a similar work. In Stability of semiclassical bound states of nonlinear Schrödinger equations with potentials, the author give a lemma as picture below
The operator $L_+^0$ is similar to $H$, but I still can't get the negative eigenvalue and kernel of $H$, and its eigenfunction respectly, since the $u_0$ in Lemma 3.1 is a given function ($u_0(x)=sqrt{2lambda}sech sqrt{2lambda }x$). In my question , the $u_0$ is not so good. I want to deal this question, what paper or book I should read ? Thanks for any help.
PS: Reference of Lemma 3.1 is
Titchmarsh, E. C., Eigenfunction expansions associated with second-order differential equations, Oxford: At the Clarendon Press. 175 p. (1946). ZBL0061.13505.
but I don't found the lemma 3.1 in it.
pde elliptic-equations
asked Nov 18 at 14:12
lanse7pty
1,7751823
add a comment |
Assume $u_0$ is a positive radial symmetric nontrival solution of
begin{align}
-A(||nabla u||^2)Delta u + lambda u - |u|^2u=0 tag{1} \
uin H^2( R^n)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
end{align}
where $A(s)$ is a given positve smooth function, $lambda >0$ is a constant. The real part of linear operator of (1) at $u_0$ is
$$
Lu=-2A'(||nabla u_0||^2) Delta u_0 int nabla u_0 cdot nabla u dx
-A(||nabla u_0||^2)Delta u +lambda u-3u_0^2 u
$$
In fact, I want to know the eigenvalue of $L$, but seemly, there is not any similar work. So, firstly, I want to know the eigenvalue of
$$
Hu=-A(||nabla u_0||^2)Delta u +lambda u-3u_0^2 u.
$$
In fact, I found a similar work. In Stability of semiclassical bound states of nonlinear Schrödinger equations with potentials, the author give a lemma as picture below
The operator $L_+^0$ is similar to $H$, but I still can't get the negative eigenvalue and kernel of $H$, and its eigenfunction respectly, since the $u_0$ in Lemma 3.1 is a given function ($u_0(x)=sqrt{2lambda}sech sqrt{2lambda }x$). In my question , the $u_0$ is not so good. I want to deal this question, what paper or book I should read ? Thanks for any help.
PS: Reference of Lemma 3.1 is
Titchmarsh, E. C., Eigenfunction expansions associated with second-order differential equations, Oxford: At the Clarendon Press. 175 p. (1946). ZBL0061.13505.
but I don't found the lemma 3.1 in it.
pde elliptic-equations
asked Nov 18 at 14:12
lanse7pty
1,7751823
Assume $u_0$ is a positive radial symmetric nontrival solution of
begin{align}
-A(||nabla u||^2)Delta u + lambda u - |u|^2u=0 tag{1} \
uin H^2( R^n)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
end{align}
where $A(s)$ is a given positve smooth function, $lambda >0$ is a constant. The real part of linear operator of (1) at $u_0$ is
$$
Lu=-2A'(||nabla u_0||^2) Delta u_0 int nabla u_0 cdot nabla u dx
-A(||nabla u_0||^2)Delta u +lambda u-3u_0^2 u
$$
In fact, I want to know the eigenvalue of $L$, but seemly, there is not any similar work. So, firstly, I want to know the eigenvalue of
$$
Hu=-A(||nabla u_0||^2)Delta u +lambda u-3u_0^2 u.
$$
In fact, I found a similar work. In Stability of semiclassical bound states of nonlinear Schrödinger equations with potentials, the author give a lemma as picture below
The operator $L_+^0$ is similar to $H$, but I still can't get the negative eigenvalue and kernel of $H$, and its eigenfunction respectly, since the $u_0$ in Lemma 3.1 is a given function ($u_0(x)=sqrt{2lambda}sech sqrt{2lambda }x$). In my question , the $u_0$ is not so good. I want to deal this question, what paper or book I should read ? Thanks for any help.
PS: Reference of Lemma 3.1 is
Titchmarsh, E. C., Eigenfunction expansions associated with second-order differential equations, Oxford: At the Clarendon Press. 175 p. (1946). ZBL0061.13505.
but I don't found the lemma 3.1 in it.
pde elliptic-equations
pde elliptic-equations
asked Nov 18 at 14:12
lanse7pty
1,7751823
asked Nov 18 at 14:12
lanse7pty
1,7751823
asked Nov 18 at 14:12
lanse7pty
1,7751823
asked Nov 18 at 14:12
lanse7pty
1,7751823
asked Nov 18 at 14:12
lanse7pty
1,7751823
1,7751823
add a comment |
add a comment |
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