On a biharmonic equations with steep potential well and indefinite potential

Abstract

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 $N\ge 3$, $a_0,b_0\in\bbr$ are two constants, $\lambda >0$ is a parameter, $b(x)\ge 0$ 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 $a_0<0$.

18 pages