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
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
MSC
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