On a biharmonic equations with steep potential well and indefinite potential

Abstract: In this paper, we study the following biharmonic equations:% $$ \left{\aligned&\Delta^2u-a_0\Delta u+(\lambda b(x)+b_0)u=f(u)&\text{ in }\bbr^N,\% &u\in\h,\endaligned\right.\eqno{(\mathcal{P}{\lambda})}% $$ where $N\geq3$, $a_0,b_0\in\bbr$ are two constants, $\lambda>0$ is a parameter, $b(x)\geq0$ is a potential well and $f(t)\in C(\bbr)$ is subcritical and superlinear or asymptotically linear at infinity. By the Gagliardo-Nirenberg inequality, we make some observations on the operator $\Delta^2-a_0\Delta+\lambda b(x)+b_0$ in $\h$. Based on these observations, we give a new variational setting to $(\mathcal{P}{\lambda})$ for $a_0<0$. With this new variational setting in hands, we establish some new existence results of the nontrivial solutions to $(\mathcal{P}{\lambda})$ for all $a_0, b_0\in\bbr$ with $\lambda$ sufficiently large by the variational method. The concentration behavior of the nontrivial solutions as $\lambda\to+\infty$ is also obtained. It is worth to point out that it seems to be the first time that the nontrivial solution of $(\mathcal{P}{\lambda})$ is obtained in the case of $a_0<0$.