一类具有陡势阱和不定势的双调和方程
On a biharmonic equations with steep potential well and indefinite potential
摘要：
本文研究了下列双调和方程：% $$ \left\{\aligned&Δ^2u-a_0Δu+(λb(x)+b_0)u=f(u)&\text{ in }\bbr^N,\\% &u\in\h,\endaligned\right.\eqno{(\mathcal{P}_λ)}% $$其中$N\geq3$，$a_0,b_0\in\bbr$是两个常数，$λ>0$是参数，$b(x)\geq0$是势阱，$f(t)\in C(\bbr)$是亚临界超线性或在无穷远处渐近线性的.利用Gagliardo-Nirenberg不等式，对$\h$中的算子$Δ^2-a_0Δ+λb(x)+b_0$进行了一些观察.基于这些观察，我们给了一个新的变分设置为$(\mathcal{P}_λ)$ $a_0<0$。利用这一新的变分方法，我们对所有$a_0, b_0\in\bbr$且$λ$充分大的情形，建立了$(\mathcal{P}_λ)$非平凡解的存在性结果.得到了非平凡解$λ\to+\infty$的浓度行为.值得指出的是，这似乎是第一次，$(\mathcal{P}_λ)$的非平凡解的情况下，$a_0<0$获得。
英文摘要：
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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