On a biharmonic equations with steep potential well and indefinite potential

In this paper, we study the following biharmonic equations:% $$ \left\{\aligned&Δ^2u-a_0Δu+(λb(x)+b_0)u=f(u)&\text{ in }\bbr^N,\\% &u\in\h,\endaligned\right.\eqno{(\mathcal{P}_λ)}% $$ where $N\geq3$, $a_0,b_0\in\bbr$ are two constants, $λ>0$ 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 $Δ^2-a_0Δ+λb(x)+b_0$ in $\h$. Based on these observations, we give a new variational setting to $(\mathcal{P}_λ)$ for $a_0<0$. With this new variational setting in hands, we establish some new existence results of the nontrivial solutions to $(\mathcal{P}_λ)$ for all $a_0, b_0\in\bbr$ with $λ$ sufficiently large by the variational method. The concentration behavior of the nontrivial solutions as $λ\to+\infty$ is also obtained. It is worth to point out that it seems to be the first time that the nontrivial solution of $(\mathcal{P}_λ)$ is obtained in the case of $a_0<0$.

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