We are concerned on the fourth-order elliptic equation \begin{equation}\tag{$P_\lambda$} \left\{ \begin{array}[c]{ll} \Delta^2 u- \Delta u + V(x)u -\lambda \Delta[\rho(u^2)]\rho'(u^2)u= f(u)\, \, \mbox{in} \, \, \mathbb{R}^N, & u\in W^{2,2}(\mathbb{R}^N), \end{array} \right. \end{equation} where $\Delta^2 = \Delta(\Delta)$ is the biharmonic operator, $3\leq N\leq 6$, the radially symmetric potential $V$ may change sign and $\inf_{\mathbb{R}^N}V(x)=-\infty$ is allowed. If $f$ satisfies a type of nonquadracity and monotonicity conditions and $\rho$ is a suitable smooth function, we prove, via variational approach, the existence of a radially symmetric nontrivial ground state solution $u_\lambda$ for problem $(P_\lambda)$ for all $\lambda\geq 0$.

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一类具有无界势的修正非线性四阶椭圆方程

我们关心的是四阶椭圆方程 \begin{equation}\tag{$P_\lambda$} \left\{ \begin{array}[c]{ll} \Delta^2 u- \Delta u + V (x)u -\lambda \Delta[\rho(u^2)]\rho'(u^2)u= f(u)\, \, \mbox{in} \, \, \mathbb{R} ^N, & u\in W^{2,2}(\mathbb{R}^N), \end{array} \right。\end{equation} 其中 $\Delta^2 = \Delta(\Delta)$ 是双调和算子，$3\leq N\leq 6$，径向对称势 $V$ 可能会改变符号和 $\inf_{\mathbb {R}^N}V(x)=-\infty$ 是允许的。如果 $f$ 满足一类非二次单调性条件并且 $\rho$ 是一个合适的光滑函数，我们通过变分方法证明问题 $(P_ \lambda)$ 为所有 $\lambda\geq 0$。