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

Volume 11, Issue 3, pp 323--334
Publication Date: February 09, 2018 Submission Date: November 29, 2017 Revision Date: December 31, 2017 Accteptance Date: January 07, 2018
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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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ISRP Style

Liuyang Shao, Haibo Chen, Ground states solutions for modified fourth-order elliptic systems with steep well potential, Journal of Nonlinear Sciences and Applications, 11 (2018), no. 3, 323--334

AMA Style

Shao Liuyang, Chen Haibo, Ground states solutions for modified fourth-order elliptic systems with steep well potential. J. Nonlinear Sci. Appl. (2018); 11(3):323--334

Chicago/Turabian Style

Shao, Liuyang, Chen, Haibo. "Ground states solutions for modified fourth-order elliptic systems with steep well potential." Journal of Nonlinear Sciences and Applications, 11, no. 3 (2018): 323--334

Keywords

• Fourth-order elliptic
• variational methods
• ground state solutions
• concentration

•  35B09
•  35J20

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