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

Shao, Liuyang¹ ; Chen, Haibo ¹

¹ School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, P. R. China

Journal of nonlinear sciences and its applications, Tome 11 (2018) no. 3, p. 323-334.

Résumé

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 $2$ $4$, $2^{\ast\ast}=\frac{2N}{N-4} \ (N\leq5)$ $(\mbox{if}~N\leq4, 2^{\ast\ast}=\infty)$ 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.

Détail

DOI : 10.22436/jnsa.011.03.01

Classification : 35B09, 35J20
Keywords: Fourth-order elliptic, variational methods, ground state solutions, concentration

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