On a Biharmonic Equation with Steep Potential Well and Indefinite Potential

Abstract

Abstract In this paper, we study the following biharmonic equations: { Δ 2 ⁢ u - a 0 ⁢ Δ ⁢ u + ( λ ⁢ b ⁢ ( x ) + b 0 ) ⁢ u = f ⁢ ( u ) in ⁢ ℝ N , u ∈ H 2 ⁢ ( ℝ N ) , $\left\{\begin{aligned} &\displaystyle\Delta^{2}u-a_{0}\Delta u+(\lambda b(x)+b% _{0})u=f(u)&\hskip 10.0pt\text{in }\mathbb{R}^{N},\\ &\displaystyle u\in H^{2}(\mathbb{R}^{N}),\end{aligned}\right.$ where N ≥ 3 ${N\geq 3}$ , a 0 , b 0 ∈ ℝ ${a_{0},b_{0}\in\mathbb{R}}$ are two constants, λ > 0 ${\lambda>0}$ is a parameter, b ⁢ ( x ) ≥ 0 ${b(x)\geq 0}$ is a potential well and f ⁢ ( t ) ∈ C ⁢ ( ℝ ) ${f(t)\in C(\mathbb{R})}$ is subcritical and superlinear or asymptotically linear at infinity. By the Gagliardo–Nirenberg inequality, we make some observations on the operator Δ 2 - a 0 ⁢ Δ + λ ⁢ b ⁢ ( x ) + b 0 ${\Delta^{2}-a_{0}\Delta+\lambda b(x)+b_{0}}$ in H 2 ⁢ ( ℝ N ) ${H^{2}(\mathbb{R}^{N})}$ . Based on these observations, we give a new variational setting to the above problem for a 0 < 0 ${a_{0}<0}$ . With this new variational setting in hands, we establish some new existence results of the nontrivial solutions for all a 0 < 0 ${a_{0}<0}$ with λ sufficiently large by the variational method. The concentration behavior of the nontrivial solution for λ → + ∞ ${\lambda\to+\infty}$ is also obtained. It is worth pointing out that it seems to be the first time that the nontrivial solution of the above problem is obtained for all a 0 < 0 ${a_{0}<0}$ .