Ground states solutions for modified fourth-order elliptic systems with steep well potential


Authors

Liuyang Shao - School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, P. R. China
Haibo Chen - School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, P. R. China


Abstract

In this paper, we study the following modified quasilinear fourth-order elliptic systems \[ \left\{\begin{array}{lll} &\triangle^{2}u-\triangle u+(\lambda\alpha(x)+1)u-\frac{1}{2}\triangle(u^{2})u=\frac{p}{p+q}|u|^{p-2}|v|^{q}u,~~&\mbox{in} \;~\mathbb{R}^{N}, \\ & \triangle^{2}v-\triangle v+(\lambda\beta(x)+1)v-\frac{1}{2}\triangle(v^{2})v=\frac{q}{p+q}|u|^{p}|v|^{q-2}v,~~&\mbox{in} \;~\mathbb{R}^{N},\end{array} \right.\] where \(\triangle^{2}=\triangle(\triangle)\) is the biharmonic operator, \(\lambda>0\), and \(2<p, 2<q,\) \(4<p+q<22^{\ast\ast}\), \(2^{\ast\ast}=\frac{2N}{N-4} \ (N\leq5)\) \((\mbox{if}~N\leq4, 2^{\ast\ast}=\infty)\) is the critical Sobolev exponent for the embedding \(W^{2,2}(\mathbb{R}^{N})\hookrightarrow L^{2^{\ast\ast}}(\mathbb{R}^{N})\). Under some appropriate assumptions on \(\alpha(x)\) and \(\beta(x)\), we obtain that the above problem has nontrivial ground state solutions via the variational methods. We also explore the phenomenon of concentration of solutions.


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