Ngô Quốc Anh

April 18, 2010

The Pohozaev identity: Elliptic problem with biharmonic operator

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 14:19

We now consider another kind of problem involving biharmonic operator. Let us assume u>0 a solution of the equation

\displaystyle (\Delta^2u)(x)+Q(x)f(u(x))=0

in \mathbb R^n. We shall prove the following result

Theorem. The following identity

\displaystyle\begin{gathered} \frac{3}{2}\left[ {\int_{{B_r}(0)}  {uQ(x)f(u)dx} - \int_{\partial {B_r}(0)} {u\frac{{\partial \Delta  u}}{{\partial \nu }}d\sigma } + \int_{\partial {B_r}(0)}  {\frac{{\partial u}}{{\partial \nu }}\Delta ud\sigma } } \right] +  \hfill \\ \frac{1}{2}\int_{\partial {B_r}(0)} {r{{(\Delta u)}^2}d\sigma }  + \int_{{B_r}(0)} {(x\cdot\nabla (\Delta u))\Delta udx} -  \int_{\partial {B_r}(0)} {\left[ {\Delta u\frac{{\partial (x\cdot\nabla  u)}}{{\partial \nu }} - (x\cdot\nabla u)\frac{{\partial \Delta  u}}{{\partial \nu }}} \right]d\sigma } \hfill \\ \qquad\qquad=  \int_{\partial {B_r}(0)} {ruQ(x)f(u)d\sigma } - \int_{{B_r}(0)} {u\left(  {Q(x)f(u) + (x\cdot\nabla Q)f(u) + (x\cdot\nabla u)Q(x)f'(u)}  \right)dx}\hfill \\ \end{gathered}

holds.

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