In this topic we consider the analysis of solutions of the following system entitled Toda system
Following is our main result
Lemma 1. The following identities
hold.
April 30, 2010
The Pohozaev identity: Toda systems and a priori estimates
![\displaystyle\begin{gathered} \left( {1 - \frac{n}{2}} \right)\int_{{B_r}(0)} {|\nabla {u_1}{|^2}dx} + r\left[ {\frac{1}{2}\int_{\partial {B_r}(0)} {|\nabla {u_1}{|^2}d\sigma } - \int_{\partial {B_r}(0)} {{{\left| {\frac{{\partial {u_1}}}{{\partial \nu }}} \right|}^2}d\sigma } } \right] \hfill \\ \qquad\qquad= - 2n\int_{{B_r}(0)} {{e^{{u_1}}}dx} + 2\int_{\partial {B_r}(0)} {r{e^{{u_1}}}dx} - \int_{{B_r}(0)} {{e^{{u_2}}}(x\cdot\nabla {u_1})dx} , \hfill \\ \left( {1 - \frac{n}{2}} \right)\int_{{B_r}(0)} {|\nabla {u_2}{|^2}dx} + r\left[ {\frac{1}{2}\int_{\partial {B_r}(0)} {|\nabla {u_2}{|^2}d\sigma } - \int_{\partial {B_r}(0)} {{{\left| {\frac{{\partial {u_2}}}{{\partial \nu }}} \right|}^2}d\sigma } } \right] \hfill \\ \qquad\qquad= - 2n\int_{{B_r}(0)} {{e^{{u_2}}}dx} + 2\int_{\partial {B_r}(0)} {r{e^{{u_2}}}dx} - \int_{{B_r}(0)} {{e^{{u_1}}}(x\cdot\nabla {u_2})dx} , \hfill \\ \end{gathered}](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Bgathered%7D+%5Cleft%28+%7B1+-+%5Cfrac%7Bn%7D%7B2%7D%7D+%5Cright%29%5Cint_%7B%7BB_r%7D%280%29%7D+%7B%7C%5Cnabla+%7Bu_1%7D%7B%7C%5E2%7Ddx%7D+%2B+r%5Cleft%5B+%7B%5Cfrac%7B1%7D%7B2%7D%5Cint_%7B%5Cpartial+%7BB_r%7D%280%29%7D+%7B%7C%5Cnabla+%7Bu_1%7D%7B%7C%5E2%7Dd%5Csigma+%7D+-+%5Cint_%7B%5Cpartial+%7BB_r%7D%280%29%7D+%7B%7B%7B%5Cleft%7C+%7B%5Cfrac%7B%7B%5Cpartial+%7Bu_1%7D%7D%7D%7B%7B%5Cpartial+%5Cnu+%7D%7D%7D+%5Cright%7C%7D%5E2%7Dd%5Csigma+%7D+%7D+%5Cright%5D+%5Chfill+%5C%5C+%5Cqquad%5Cqquad%3D+-+2n%5Cint_%7B%7BB_r%7D%280%29%7D+%7B%7Be%5E%7B%7Bu_1%7D%7D%7Ddx%7D+%2B+2%5Cint_%7B%5Cpartial+%7BB_r%7D%280%29%7D+%7Br%7Be%5E%7B%7Bu_1%7D%7D%7Ddx%7D+-+%5Cint_%7B%7BB_r%7D%280%29%7D+%7B%7Be%5E%7B%7Bu_2%7D%7D%7D%28x%5Ccdot%5Cnabla+%7Bu_1%7D%29dx%7D+%2C+%5Chfill+%5C%5C+%5Cleft%28+%7B1+-+%5Cfrac%7Bn%7D%7B2%7D%7D+%5Cright%29%5Cint_%7B%7BB_r%7D%280%29%7D+%7B%7C%5Cnabla+%7Bu_2%7D%7B%7C%5E2%7Ddx%7D+%2B+r%5Cleft%5B+%7B%5Cfrac%7B1%7D%7B2%7D%5Cint_%7B%5Cpartial+%7BB_r%7D%280%29%7D+%7B%7C%5Cnabla+%7Bu_2%7D%7B%7C%5E2%7Dd%5Csigma+%7D+-+%5Cint_%7B%5Cpartial+%7BB_r%7D%280%29%7D+%7B%7B%7B%5Cleft%7C+%7B%5Cfrac%7B%7B%5Cpartial+%7Bu_2%7D%7D%7D%7B%7B%5Cpartial+%5Cnu+%7D%7D%7D+%5Cright%7C%7D%5E2%7Dd%5Csigma+%7D+%7D+%5Cright%5D+%5Chfill+%5C%5C+%5Cqquad%5Cqquad%3D+-+2n%5Cint_%7B%7BB_r%7D%280%29%7D+%7B%7Be%5E%7B%7Bu_2%7D%7D%7Ddx%7D+%2B+2%5Cint_%7B%5Cpartial+%7BB_r%7D%280%29%7D+%7Br%7Be%5E%7B%7Bu_2%7D%7D%7Ddx%7D+-+%5Cint_%7B%7BB_r%7D%280%29%7D+%7B%7Be%5E%7B%7Bu_1%7D%7D%7D%28x%5Ccdot%5Cnabla+%7Bu_2%7D%29dx%7D+%2C+%5Chfill+%5C%5C+%5Cend%7Bgathered%7D&bg=ffffff&fg=333333&s=0 "\displaystyle\begin{gathered} \left( {1 - \frac{n}{2}} \right)\int_{{B_r}(0)} {|\nabla {u_1}{|^2}dx} + r\left[ {\frac{1}{2}\int_{\partial {B_r}(0)} {|\nabla {u_1}{|^2}d\sigma } - \int_{\partial {B_r}(0)} {{{\left| {\frac{{\partial {u_1}}}{{\partial \nu }}} \right|}^2}d\sigma } } \right] \hfill \\ \qquad\qquad= - 2n\int_{{B_r}(0)} {{e^{{u_1}}}dx} + 2\int_{\partial {B_r}(0)} {r{e^{{u_1}}}dx} - \int_{{B_r}(0)} {{e^{{u_2}}}(x\cdot\nabla {u_1})dx} , \hfill \\ \left( {1 - \frac{n}{2}} \right)\int_{{B_r}(0)} {|\nabla {u_2}{|^2}dx} + r\left[ {\frac{1}{2}\int_{\partial {B_r}(0)} {|\nabla {u_2}{|^2}d\sigma } - \int_{\partial {B_r}(0)} {{{\left| {\frac{{\partial {u_2}}}{{\partial \nu }}} \right|}^2}d\sigma } } \right] \hfill \\ \qquad\qquad= - 2n\int_{{B_r}(0)} {{e^{{u_2}}}dx} + 2\int_{\partial {B_r}(0)} {r{e^{{u_2}}}dx} - \int_{{B_r}(0)} {{e^{{u_1}}}(x\cdot\nabla {u_2})dx} , \hfill \\ \end{gathered}")
April 18, 2010
The Pohozaev identity: Elliptic problem with biharmonic operator
We now consider another kind of problem involving biharmonic operator. Let us assume 
in 
Theorem. The following identity
holds.
![\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}](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Bgathered%7D+%5Cfrac%7B3%7D%7B2%7D%5Cleft%5B+%7B%5Cint_%7B%7BB_r%7D%280%29%7D++%7BuQ%28x%29f%28u%29dx%7D+-+%5Cint_%7B%5Cpartial+%7BB_r%7D%280%29%7D+%7Bu%5Cfrac%7B%7B%5Cpartial+%5CDelta++u%7D%7D%7B%7B%5Cpartial+%5Cnu+%7D%7Dd%5Csigma+%7D+%2B+%5Cint_%7B%5Cpartial+%7BB_r%7D%280%29%7D++%7B%5Cfrac%7B%7B%5Cpartial+u%7D%7D%7B%7B%5Cpartial+%5Cnu+%7D%7D%5CDelta+ud%5Csigma+%7D+%7D+%5Cright%5D+%2B++%5Chfill+%5C%5C+%5Cfrac%7B1%7D%7B2%7D%5Cint_%7B%5Cpartial+%7BB_r%7D%280%29%7D+%7Br%7B%7B%28%5CDelta+u%29%7D%5E2%7Dd%5Csigma+%7D++%2B+%5Cint_%7B%7BB_r%7D%280%29%7D+%7B%28x%5Ccdot%5Cnabla+%28%5CDelta+u%29%29%5CDelta+udx%7D+-++%5Cint_%7B%5Cpartial+%7BB_r%7D%280%29%7D+%7B%5Cleft%5B+%7B%5CDelta+u%5Cfrac%7B%7B%5Cpartial+%28x%5Ccdot%5Cnabla++u%29%7D%7D%7B%7B%5Cpartial+%5Cnu+%7D%7D+-+%28x%5Ccdot%5Cnabla+u%29%5Cfrac%7B%7B%5Cpartial+%5CDelta++u%7D%7D%7B%7B%5Cpartial+%5Cnu+%7D%7D%7D+%5Cright%5Dd%5Csigma+%7D+%5Chfill+%5C%5C+%5Cqquad%5Cqquad%3D++%5Cint_%7B%5Cpartial+%7BB_r%7D%280%29%7D+%7BruQ%28x%29f%28u%29d%5Csigma+%7D+-+%5Cint_%7B%7BB_r%7D%280%29%7D+%7Bu%5Cleft%28++%7BQ%28x%29f%28u%29+%2B+%28x%5Ccdot%5Cnabla+Q%29f%28u%29+%2B+%28x%5Ccdot%5Cnabla+u%29Q%28x%29f%27%28u%29%7D++%5Cright%29dx%7D%5Chfill+%5C%5C+%5Cend%7Bgathered%7D&bg=ffffff&fg=333333&s=0 "\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}")
April 14, 2010
The Pohozaev identity: Integral equation with exponential nonlinearity
We now consider the Pohozaev identity for some integral equations. We start with the following equation
where 
Theorem. Suppose
is a
solution of the above integral equation such that
is absolutiely integrable over
then
and the following identity holds
.
This theorem was due to Xu X.W. from the paper published in J. Funct. Anal. (2005). When 
Finiteness for 
April 13, 2010
The Pohozaev identity: Semilinear elliptic problem with exponential nonlinearity
We know consider another type of equation, precisely, we consider the positive solution to the following
in
Theorem. The following identity
holds.
![\displaystyle\begin{gathered} \left( {1 + \frac{n}{2}} \right)\int_{{B_r}(0)} {uK{e^{2u}}dx} + \int_{{B_r}(0)} {u\left( {(x \cdot \nabla K){e^{2u}} + 2(x \cdot \nabla u)K{e^{2u}}} \right)dx} \hfill \\ \qquad= \int_{\partial {B_r}(0)} {ruK{e^{2u}}d\sigma } - \int_{\partial {B_r}(0)} {\left[ {\left( {1 - \frac{n}{2}} \right)u\frac{{\partial u}}{{\partial \nu }} + \frac{1}{2}r|\nabla u{|^2} - r{{\left| {\frac{{\partial u}}{{\partial \nu }}} \right|}^2}} \right]d\sigma } \hfill \\ \end{gathered}](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Bgathered%7D+%5Cleft%28+%7B1+%2B+%5Cfrac%7Bn%7D%7B2%7D%7D++%5Cright%29%5Cint_%7B%7BB_r%7D%280%29%7D+%7BuK%7Be%5E%7B2u%7D%7Ddx%7D+%2B+%5Cint_%7B%7BB_r%7D%280%29%7D+%7Bu%5Cleft%28+%7B%28x++%5Ccdot+%5Cnabla+K%29%7Be%5E%7B2u%7D%7D+%2B+2%28x+%5Ccdot+%5Cnabla+u%29K%7Be%5E%7B2u%7D%7D%7D+%5Cright%29dx%7D++%5Chfill+%5C%5C+%5Cqquad%3D+%5Cint_%7B%5Cpartial+%7BB_r%7D%280%29%7D+%7BruK%7Be%5E%7B2u%7D%7Dd%5Csigma+%7D+-++%5Cint_%7B%5Cpartial+%7BB_r%7D%280%29%7D+%7B%5Cleft%5B+%7B%5Cleft%28+%7B1+-+%5Cfrac%7Bn%7D%7B2%7D%7D++%5Cright%29u%5Cfrac%7B%7B%5Cpartial+u%7D%7D%7B%7B%5Cpartial+%5Cnu+%7D%7D+%2B+%5Cfrac%7B1%7D%7B2%7Dr%7C%5Cnabla++u%7B%7C%5E2%7D+-+r%7B%7B%5Cleft%7C+%7B%5Cfrac%7B%7B%5Cpartial+u%7D%7D%7B%7B%5Cpartial+%5Cnu+%7D%7D%7D+%5Cright%7C%7D%5E2%7D%7D++%5Cright%5Dd%5Csigma+%7D+%5Chfill+%5C%5C+%5Cend%7Bgathered%7D&bg=ffffff&fg=333333&s=0 "\displaystyle\begin{gathered} \left( {1 + \frac{n}{2}} \right)\int_{{B_r}(0)} {uK{e^{2u}}dx} + \int_{{B_r}(0)} {u\left( {(x \cdot \nabla K){e^{2u}} + 2(x \cdot \nabla u)K{e^{2u}}} \right)dx} \hfill \\ \qquad= \int_{\partial {B_r}(0)} {ruK{e^{2u}}d\sigma } - \int_{\partial {B_r}(0)} {\left[ {\left( {1 - \frac{n}{2}} \right)u\frac{{\partial u}}{{\partial \nu }} + \frac{1}{2}r|\nabla u{|^2} - r{{\left| {\frac{{\partial u}}{{\partial \nu }}} \right|}^2}} \right]d\sigma } \hfill \\ \end{gathered}")
April 11, 2010
The Pohozaev identity: Semilinear elliptic problem with polygonal nonlinearity
Let us start with the Pohozaev identity for semilinear elliptic equation with polygonal nonlinear terms of the form
over an open, star-shaped domain 
We multiply the PDE by 
The term on the left is just
![\displaystyle\begin{gathered} A = - \sum\limits_{j = 1}^n {\sum\limits_{i = 1}^n {\int_\Omega {{u_{{x_i}{x_i}}}{x_j}{u_{{x_j}}}dx} } } \hfill \\ \quad= - \sum\limits_{j = 1}^n {\sum\limits_{i = 1}^n {\left[ { - \int_\Omega {{u_{{x_i}}}{{({x_j}u)}_{{x_i}}}dx} + \int_{\partial \Omega } {{u_{{x_i}}}{\nu ^i}{x_j}{u_{{x_j}}}d\sigma } } \right]} } \hfill \\ \quad= \underbrace {\sum\limits_{j = 1}^n {\sum\limits_{i = 1}^n {\int_\Omega {{u_{{x_i}}}{{({x_j}u_{x_j})}_{{x_i}}}dx} } } }_{{A_1}} - \underbrace {\sum\limits_{j = 1}^n {\sum\limits_{i = 1}^n {\int_{\partial \Omega } {{u_{{x_i}}}{\nu ^i}{x_j}{u_{{x_j}}}d\sigma } } } }_{{A_2}}. \hfill \\ \end{gathered}](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Bgathered%7D+A+%3D+-+%5Csum%5Climits_%7Bj+%3D+1%7D%5En+%7B%5Csum%5Climits_%7Bi+%3D+1%7D%5En+%7B%5Cint_%5COmega+%7B%7Bu_%7B%7Bx_i%7D%7Bx_i%7D%7D%7D%7Bx_j%7D%7Bu_%7B%7Bx_j%7D%7D%7Ddx%7D+%7D+%7D+%5Chfill+%5C%5C+%5Cquad%3D+-+%5Csum%5Climits_%7Bj+%3D+1%7D%5En+%7B%5Csum%5Climits_%7Bi+%3D+1%7D%5En+%7B%5Cleft%5B+%7B+-+%5Cint_%5COmega+%7B%7Bu_%7B%7Bx_i%7D%7D%7D%7B%7B%28%7Bx_j%7Du%29%7D_%7B%7Bx_i%7D%7D%7Ddx%7D+%2B+%5Cint_%7B%5Cpartial+%5COmega+%7D+%7B%7Bu_%7B%7Bx_i%7D%7D%7D%7B%5Cnu+%5Ei%7D%7Bx_j%7D%7Bu_%7B%7Bx_j%7D%7D%7Dd%5Csigma+%7D+%7D+%5Cright%5D%7D+%7D+%5Chfill+%5C%5C+%5Cquad%3D+%5Cunderbrace+%7B%5Csum%5Climits_%7Bj+%3D+1%7D%5En+%7B%5Csum%5Climits_%7Bi+%3D+1%7D%5En+%7B%5Cint_%5COmega+%7B%7Bu_%7B%7Bx_i%7D%7D%7D%7B%7B%28%7Bx_j%7Du_%7Bx_j%7D%29%7D_%7B%7Bx_i%7D%7D%7Ddx%7D+%7D+%7D+%7D_%7B%7BA_1%7D%7D+-+%5Cunderbrace+%7B%5Csum%5Climits_%7Bj+%3D+1%7D%5En+%7B%5Csum%5Climits_%7Bi+%3D+1%7D%5En+%7B%5Cint_%7B%5Cpartial+%5COmega+%7D+%7B%7Bu_%7B%7Bx_i%7D%7D%7D%7B%5Cnu+%5Ei%7D%7Bx_j%7D%7Bu_%7B%7Bx_j%7D%7D%7Dd%5Csigma+%7D+%7D+%7D+%7D_%7B%7BA_2%7D%7D.+%5Chfill+%5C%5C+%5Cend%7Bgathered%7D&bg=ffffff&fg=333333&s=0 "\displaystyle\begin{gathered} A = - \sum\limits_{j = 1}^n {\sum\limits_{i = 1}^n {\int_\Omega {{u_{{x_i}{x_i}}}{x_j}{u_{{x_j}}}dx} } } \hfill \\ \quad= - \sum\limits_{j = 1}^n {\sum\limits_{i = 1}^n {\left[ { - \int_\Omega {{u_{{x_i}}}{{({x_j}u)}_{{x_i}}}dx} + \int_{\partial \Omega } {{u_{{x_i}}}{\nu ^i}{x_j}{u_{{x_j}}}d\sigma } } \right]} } \hfill \\ \quad= \underbrace {\sum\limits_{j = 1}^n {\sum\limits_{i = 1}^n {\int_\Omega {{u_{{x_i}}}{{({x_j}u_{x_j})}_{{x_i}}}dx} } } }_{{A_1}} - \underbrace {\sum\limits_{j = 1}^n {\sum\limits_{i = 1}^n {\int_{\partial \Omega } {{u_{{x_i}}}{\nu ^i}{x_j}{u_{{x_j}}}d\sigma } } } }_{{A_2}}. \hfill \\ \end{gathered}")
