The Existence of a Nontrivial Solution for Biharmonic Equation
重调和方程非平凡解的存在性

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.