Abstract:

The paper mainly studies biharmonic equation in $R^N(N>4)$ as
$$
\left\{
\begin{array}{ll}
\Delta^2 u+\lambda u=\overline{f}(x,u);\\
\lim\limits_{|x|\rightarrow\infty}u(x)=0;\\
u\in{H^2}(R^N),\hspace{0.1cm}x\in{R^N }.
\end{array}
\right.
$$

For studying it, the authors change it to the biharmonic equation with a perturbation in $R^N(N>4)$ as
$$
\left\{
\begin{array}{ll}
\Delta^2 u+\lambda u=f(u)+\varepsilon g(x,u);\\
\lim\limits_{|x|\rightarrow\infty}u(x)=0;\\
u\in{H^2}(R^N),\hspace{0.1cm}x\in{R^N }
\end{array}
\right.
$$
and use the perturbation method to study it (where
$f(u)=\lim\limits_{|x|\longrightarrow \infty}\overline{f}(x,u),\varepsilon g(x,u)=\overline{f}(x,u)-f(u),\varepsilon$
is a small constant).

The authors can prove the existence of nontrivial
solutions of the above question under some conditions.

Key words: Existence, Biharmonic equation, Perturbative