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

Volume 11, Issue 3, pp 323--334 http://dx.doi.org/10.22436/jnsa.011.03.01 Publication Date: February 09, 2018       Article History

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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