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$L^2$-norm of the solution of a 2-order elliptic nonlinear equation

0

$begingroup$

When $1<p<2^*$, $E>0$ is a constant, assuming the positive solution of equation below is $U_E$
$$
-Delta u + Eu - |u|^p-1u =0, ~~~~xin mathbb R^n (nge1)
$$
as I read in Evans' book, the term $Eu$ can be treat as heat source, and the solution is the heat distribution, so I think $E$ is bigger, the $||U_E||_L^2$ is bigger. But I don't know how to prove it, any way to start it. Obviously,
$$
fracddE||U_E||_L^2^2 =fracddEleft[-frac1Eint |nabla U_E|^2+frac1Eint |U_E|^p+1right]
$$
but the right part is still hard to know whether it is positive, how should I do ?

pde elliptic-equations

share|cite|improve this question

asked Apr 2 at 8:16

lanse7ptylanse7pty

1,8711823

$endgroup$

add a comment | 

0

$begingroup$

When $1<p<2^*$, $E>0$ is a constant, assuming the positive solution of equation below is $U_E$
$$
-Delta u + Eu - |u|^p-1u =0, ~~~~xin mathbb R^n (nge1)
$$
as I read in Evans' book, the term $Eu$ can be treat as heat source, and the solution is the heat distribution, so I think $E$ is bigger, the $||U_E||_L^2$ is bigger. But I don't know how to prove it, any way to start it. Obviously,
$$
fracddE||U_E||_L^2^2 =fracddEleft[-frac1Eint |nabla U_E|^2+frac1Eint |U_E|^p+1right]
$$
but the right part is still hard to know whether it is positive, how should I do ?

pde elliptic-equations

share|cite|improve this question

asked Apr 2 at 8:16

lanse7ptylanse7pty

1,8711823

$endgroup$

add a comment | 

$begingroup$

When $1<p<2^*$, $E>0$ is a constant, assuming the positive solution of equation below is $U_E$
$$
-Delta u + Eu - |u|^p-1u =0, ~~~~xin mathbb R^n (nge1)
$$
as I read in Evans' book, the term $Eu$ can be treat as heat source, and the solution is the heat distribution, so I think $E$ is bigger, the $||U_E||_L^2$ is bigger. But I don't know how to prove it, any way to start it. Obviously,
$$
fracddE||U_E||_L^2^2 =fracddEleft[-frac1Eint |nabla U_E|^2+frac1Eint |U_E|^p+1right]
$$
but the right part is still hard to know whether it is positive, how should I do ?

pde elliptic-equations

share|cite|improve this question

asked Apr 2 at 8:16

lanse7ptylanse7pty

1,8711823

$endgroup$

share|cite|improve this question

asked Apr 2 at 8:16

lanse7ptylanse7pty

1,8711823

share|cite|improve this question

asked Apr 2 at 8:16

lanse7ptylanse7pty

1,8711823

asked Apr 2 at 8:16

lanse7ptylanse7pty

1,8711823

asked Apr 2 at 8:16

lanse7ptylanse7pty

1,8711823