In this paper, we study the existence of multiple solutions for the following biharmonic problem

$$\begin{aligned} \Delta ^2 u= & {} f(x,u) + g(x,u)\quad \hbox {in}\quad \Omega ,\\ u= & {} \Delta u =0 \quad \hbox {on}\quad \partial \Omega , \end{aligned}$$

where \(\Omega \subset {\mathbb {R}}^N, (N > 4)\) is a smooth bounded domain and \(f(x,\xi )\) is odd in \(\xi , g(x,\xi )\) is a perturbation term. By using the variant of Rabinowitz’s perturbation method, under some growth conditions on f and g, we show that there are infinitely many weak solutions to the problem.

中文翻译：

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受对称性扰动的四阶半线性椭圆方程的无穷多个解

在本文中，我们研究以下双谐波问题的多重解的存在

$$ \ begin {aligned} \ Delta ^ 2 u =＆{} f（x，u）+ g（x，u）\ quad \ hbox {in} \ quad \ Omega，\\ u =＆{} \ Delta u = 0 \ quad \ hbox {on} \ quad \ partial \ Omega，\ end {aligned} $$

其中\（\ Omega \ subset {\ mathbb {R}} ^ N，（N> 4）\）是一个光滑有界域，\（f（x，\ xi）\）在\（\ xi，g （x，\ xi）\）是一个扰动项。通过使用拉比诺维茨摄动法的变体，在f和g上的某些增长条件下，我们证明了存在无限多个弱解。